JEE Advanced Maths PYQ – Conic Sections (Circles) | Previous Year Questions with Solutions

Q 1 :

Consider a triangle $Δ$ whose two sides lie on the $x$ -axis and the line $x + y + 1 = 0$ . If the orthocenter of $Δ$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $Δ$ is [2021]

$x^{2} + y^{2} - 3 x + y = 0$
$x^{2} + y^{2} + x + 3 y = 0$
$x^{2} + y^{2} + 2 y - 1 = 0$
$x^{2} + y^{2} + x + y = 0$

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(2)

[IMAGE 300]

$(1, - 2) = (α, - α - 1) \Rightarrow α = 1$

It is clear from question that one of the vertex of triangle is intersection of $x$ -axis and $x + y + 1 = 0 \Rightarrow A (- 1, 0)$

$Let vertex B be (α, - α - 1)$

$Line A C ⊥ B H, so m_{A C} \cdot m_{B H} = - 1$

$\Rightarrow 0 = - \frac{(1 - α)}{α + 2} \Rightarrow α = 1 \Rightarrow B (1, - 2)$

$Let vertex C be (β, 0)$

$Line A H ⊥ B C$

$∴$ $m_{A H} \cdot m_{B C} = - 1$

$\Rightarrow \frac{1}{2} \cdot \frac{2}{β - 1} = - 1 \Rightarrow β = 0$

$Centroid of ∆ A B C is (0, - \frac{2}{3})$

$We know that G (centroid) divides line joining circumcentre (O) and orthocentre (H) in the ratio 1 : 2$

[IMAGE 301]

$\Rightarrow 2 h + 1 = 0$ $\Rightarrow \frac{2 k + 1}{3} = - \frac{2}{3}$

$\Rightarrow h = - \frac{1}{2}$ $\Rightarrow k = - \frac{3}{2}$ $\Rightarrow Circumcentre is (- \frac{1}{2}, - \frac{3}{2})$

$Equation of circumcircle is (passing hrough C (0, 0)) is$ $x^{2} + y^{2} + x + 3 y = 0$

Q 2 :

A line $y = m x + 1$ intersects the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ at the points P and Q. If the midpoint of the line segment PQ has $x$ -coordinate $- \frac{3}{5}$ , then which one of the following options is correct? [2019]

$2 \leq m < 4$
$- 3 \leq m < - 1$
$4 \leq m < 6$
$6 \leq m < 8$

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(1)

Given: Circle ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ , with centre $C (3, - 2)$ and radius 5 is intersected by a line $y = m x + 1$ at P & Q such that coordinates of midpoint R of PQ is $- \frac{3}{5}$ .

Since $x$ -coordinate of point R is $- \frac{3}{5}$ and point R lies on the line $y = m x + 1$ , therefore $y$ -coordinate of R will be $\frac{3 m}{5} + 1$ .

[IMAGE 302]

$∴$ $R (- \frac{3}{5}, - \frac{3 m}{5} + 1)$

Since R is the midpoint of PQ, therefore $C R ⊥ P Q$

$\Rightarrow \frac{- \frac{3 m}{5} + 1 + 2}{- \frac{3}{5} - 3} \times m = - 1$

$\Rightarrow m^{2} - 5 m + 6 = 0 \Rightarrow m = 2, 3$

Q 3 :

The circle passing through the point $(- 1, 0)$ and touching the $y$ -axis at (0, 2) also passes through the point. [2011]

$(- \frac{3}{2}, 0)$
$(- \frac{5}{2}, 2)$
$(- \frac{3}{2}, \frac{5}{2})$
$(- 4, 0)$

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(4)

Let centre of the circle be $(h, 2)$ , then radius = $h$

$∴$ Equation of circle becomes ${(x - h)}^{2} + {(y - 2)}^{2} = h^{2}$

Since the circle passes through $(- 1, 0)$

[IMAGE 303]

$∴$ ${(- 1 - h)}^{2} + 4 = h^{2} \Rightarrow h = \frac{- 5}{2}$

$∴$ $Centre (- \frac{5}{2}, 2) and r = \frac{5}{2}$

$Now, distance of centre from (- 4, 0) is \frac{5}{2}$

$∴$ $Point (- 4, 0) lies on the circle.$

Q 4 :

Tangents drawn from the point P(1, 8) to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the points A and B. The equation of the circumcircle of the triangle $P A B$ is [2009]

$x^{2} + y^{2} + 4 x - 6 y + 19 = 0$
$x^{2} + y^{2} - 4 x - 10 y + 19 = 0$
$x^{2} + y^{2} - 2 x + 6 y - 29 = 0$
$x^{2} + y^{2} - 6 x - 4 y + 19 = 0$

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(2)

Given that tangents PA and PB are drawn from the point P(1, 3) to circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ with centre C(3, 2)

[IMAGE 304]

Clearly the circumcircle of $∆ P A B$ will pass through C and as $∠ A = 90 °$ , PC must be a diameter of the circle.

$∴$ Equation of required circle is

$(x - 1) (x - 3) + (y - 8) (y - 2) = 0$

$\Rightarrow x^{2} + y^{2} - 4 x - 10 y + 19 = 0$

Q 5 :

If one of the diameters of the circle $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ is a chord to the circle with centre (2, 1), then the radius of the circle is [2004]

$\sqrt{3}$
$\sqrt{2}$
$3$
$2$

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(3)

The given circle is $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ with centre C(1, 3) and radius $= \sqrt{1 + 9 - 6} = 2$ .

Let AB be one of its diameter which is the chord of other circle with centre at $C_{1} (2, 1)$ .

[IMAGE 305]

$Then in ∆ C_{1} C B,$

$C_{1} B^{2} = C C_{1}^{2} + C B^{2}$

$= {(2 - 1)}^{2} + {(1 - 3)}^{2} + {(2)}^{2}$

$= 1 + 4 + 4 = 9 \Rightarrow C_{1} B = 3$

Q 6 :

The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]

3
6
9
15

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(4)

[IMAGE 306]

$Let the tangent to y^{2} = 8 x be y = m x + \frac{2}{m}$

If it is common tangent to parabola and circle $x^{2} + y^{2} = 2$ , then distance of the tangent from the centre of the circle is equal to radius of the circle

$∴$ $| \frac{\frac{2}{m}}{\sqrt{m^{2} + 1}} |$ $= \sqrt{2} \Rightarrow \frac{4}{m^{2} (1 + m^{2})} = 2$

$\Rightarrow m^{4} + m^{2} - 2 = 0 \Rightarrow (m^{2} + 2) (m^{2} - 1) = 0 \Rightarrow m = 1 or - 1$

$∴$ $Required tangents are y = x + 2 and y = - x - 2$

$Their common point is (- 2, 0)$

$∴$ $Tangents are drawn from (- 2, 0)$

$∴$ $Chord of contact P Q to circle is$ $x . (- 2) + y . (0) = 2 \Rightarrow x = - 1$

$and chord of contact R S to parabola is$ $y (0) = 4 (x - 2) \Rightarrow x = 2$

$Hence coordinates of P and Q are (- 1, 1) and (- 1, - 1) respectively.$

$Also coordinates of R and S are (2, - 4) and (2, 4) respectively.$

$∴$ $Area of trapezium P Q R S = \frac{1}{2} (2 + 8) \times 3 = 15$

Q 7 :

A circle is given by $x^{2} + {(y - 1)}^{2} = 1$ , another circle $C$ touches it externally and also the $x$ -axis, then the locus of its centre is [2005]

${(x, y) : x^{2} = 4 y} \cup {(x, y) : y \leq 0}$
${(x, y) : x^{2} + {(y - 1)}^{2} = 4} \cup {(x, y) : y \leq 0}$
${(x, y) : x^{2} = y} \cup {(0, y) : y \leq 0}$
${(x, y) : x^{2} = 4 y} \cup {(0, y) : y \leq 0}$

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(4)

Let the centre of circle $C$ be $(h, k)$ . This circle touches $x$ -axis.

$∴$ $radius =$ $| k |$

[IMAGE 307]

Also it touches the given circle $x^{2} + {(y - 1)}^{2} = 1$ , with centre (0, 1) and radius 1, externally.

$∴$ Distance between centres = sum of radii

$\Rightarrow \sqrt{{(h - 0)}^{2} + {(k - 1)}^{2}} = 1 +$ $| k |$

$\Rightarrow h^{2} + k^{2} - 2 k + 1 = 1 + 2$ $| k |$ $+ k^{2}$

$\Rightarrow h^{2} = 2 k + 2$ $| k |$

$∴$ $Locus of (h, k) is x^{2} = 2 y + 2$ $| y |$

$Now if y > 0, it becomes x^{2} = 4 y$

$and if y \leq 0, it becomes x = 0$

$∴$ $Combining the two, the required locus is {(x, y) : x^{2} = 4 y} \cup {(0, y) : y \leq 0}$

Q 8 :

The centre of circle inscribed in square formed by the lines $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ , is [2003]

(4, 7)
(7, 4)
(9, 4)
(4, 9)

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(1)

$x^{2} - 8 x + 12 = 0 \Rightarrow (x - 6) (x - 2) = 0$

$y^{2} - 14 y + 45 = 0 \Rightarrow (y - 5) (y - 9) = 0$

$Hence, sides of square are x = 2, x = 6, y = 5 and y = 9$

Therefore, centre of circle inscribed in square will be

$(\frac{2 + 6}{2}, \frac{5 + 9}{2}) = (4, 7)$

Q 9 :

If the tangent at the point $P$ on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets a straight line $5 x - 2 y + 6 = 0$ at a point $Q$ on the $y$ -axis, then the length of $P Q$ is [2002]

$4$
$2 \sqrt{5}$
$5$
$3 \sqrt{5}$

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(3)

Given that line $5 x - 2 y + 6 = 0$ is intersected by tangent at $P$ to the circle $x^{2} + y^{2} + 6 x + 6 y - 2 = 0$ on $y$ -axis at $Q (0, 3)$ .

This means that tangent passes through (0, 3)

$∴$ $P Q = length of tangent to circle from (0, 3)$

$= \sqrt{0^{2} + 3^{2} + 6 (0) + 6 (3) - 2} = \sqrt{0 + 9 + 0 + 18 - 2} = \sqrt{25} = 5$

Q 10 :

Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$ . If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals [2001]

$\sqrt{P Q \cdot R S}$
$\frac{P Q + R S}{2}$
$\frac{2 P Q \cdot R S}{P Q + R S}$
$\frac{\sqrt{(P Q^{2} + R S^{2})}}{2}$

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(1)

$Let ∠ R P S = θ,$ $∴$ $∠ X P Q = 90 ° - θ$

[IMAGE 308]

$and ∠ P Q X = θ$ $(∵ ∠ P X Q = 90 °)$

$∴$ $∆ P R S ~ ∆ Q P R$ $(By AA similarity)$

$∴$ $\frac{P R}{Q P} = \frac{R S}{P R} \Rightarrow P R^{2} = P Q \cdot R S$

$\Rightarrow P R = \sqrt{P Q \cdot R S} \Rightarrow 2 r = \sqrt{P Q \cdot R S}$

Q 11 :

Let $A B$ be a chord of the circle $x^{2} + y^{2} = r^{2}$ subtending a right angle at the centre. Then the locus of the centroid of the triangle $P A B$ as $P$ moves on the circle is [2001]

a parabola
a circle
an ellipse
a pair of straight lines

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$Given a circle x^{2} + y^{2} = r^{2} with centre at (0, 0) and radius r .$

[IMAGE 309]

$Let A and B be (- r, 0) and (0, - r), so that ∠ A O B = 90 °$

$and an arbitrary point P on the given circle be (r \cos θ, r \sin θ) .$

$For locus of centroid of ∆ A B P$

$(\frac{r \cos θ - r}{3}, \frac{r \sin θ - r}{3}) = (x, y)$

$\Rightarrow r \cos θ - r = 3 x, r \sin θ - r = 3 y$

$\Rightarrow r \cos θ = 3 x + r, r \sin θ = 3 y + r$

$On squaring and adding,$

${(3 x + r)}^{2} + {(3 y + r)}^{2} = r^{2},$ $which is a circle.$

Q 12 :

If the circles $x^{2} + y^{2} + 2 x + 2 k y + 6 = 0$ , $x^{2} + y^{2} + 2 k y + k = 0$ intersect orthogonally, then $k$ is [2000]

$2 or - \frac{3}{2}$
$- 2 or - \frac{3}{2}$
$2 or \frac{3}{2}$
$- 2 or \frac{3}{2}$

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(1)

Two circles intersect each other orthogonally if $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Since the two given circles intersect each other orthogonally,

$∴$ $2 (1) (0) + 2 (k) (k) = 6 + k$

$\Rightarrow 2 k^{2} - k - 6 = 0$

$\Rightarrow k = - \frac{3}{2}, 2$

Q 13 :

The triangle $P Q R$ is inscribed in the circle $x^{2} + y^{2} = 25$ . If $Q$ and $R$ have co-ordinates (3, 4) and $(- 4, 3)$ respectively, then $∠ Q P R$ is equal to [2000]

$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{4}$
$\frac{π}{6}$

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(3)

$O$ is the point at centre and $P$ is the point at circumference. Therefore, angle QOR is double the angle QPR.

[IMAGE 310]

So, it is sufficient to find the angle QOR. Now slope of $O Q = \frac{4}{3} = m_{1} (let)$

Slope of $O R = - \frac{3}{4} = m_{2} (let); Now, m_{1} m_{2} = - 1$

$Therefore, ∠ Q O R = 90 °$ $∴$ $∠ Q P R = 45 ° .$

Q 14 :

Let $A_{1}, A_{2}, A_{3}, \dots, A_{8}$ be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let $P A_{i}$ denote the distance between the points P and $A_{i}$ for $i = 1, 2, \dots, 8$ . If P varies over the circle, then the maximum value of the product $P A_{1} \cdot P A_{2} \dots P A_{8}$ is [2023]

(512)

[IMAGE 311]

$In ∆ A_{1} O P, ∠ O A_{1} P = ∠ O P A_{1} = \frac{π}{2} - \frac{θ}{2}$

$\frac{P A_{1}}{2} = \frac{\sin θ}{\sin (\frac{π}{2} - \frac{θ}{2})} = 2 \sin \frac{θ}{2} \Rightarrow P A_{1} = 4 \sin (\frac{θ}{2}) = x_{1}$ (say)

$P A_{8} = 4 \sin (\frac{π}{8} + \frac{θ}{2}) = x_{8}$ $[∵ ∠ P A_{8} O = \frac{π}{8} + \frac{θ}{2}]$

$P A_{7} = 4 \sin (\frac{π}{4} + \frac{θ}{2}) = x_{7}, P A_{6} = 4 \sin (\frac{3 π}{8} + \frac{θ}{2}) = x_{6}$

Similarly

$P A_{2} = 4 \sin (\frac{ϕ}{2}) = x_{2}, P A_{3} = 4 \sin (\frac{π}{8} + \frac{ϕ}{2}) = x_{3}$

$P A_{4} = 4 \sin (\frac{π}{4} + \frac{ϕ}{2}) = x_{4}, P A_{5} = 4 \sin (\frac{3 π}{8} + \frac{ϕ}{2}) = x_{5}$

$Now, P A_{1} \cdot P A_{2} \dots P A_{8} =$

$P = 4^{8} \sin (\frac{θ}{2}) \sin (\frac{3 π}{8} + \frac{ϕ}{2}) . \sin (\frac{π}{8} + \frac{θ}{2}) \sin (\frac{π}{4} + \frac{θ}{2})$ $\sin (\frac{π}{4} + \frac{ϕ}{2}) \sin (\frac{π}{8} + \frac{ϕ}{2}) . \sin (\frac{3 π}{8} + \frac{θ}{2}) \sin (\frac{ϕ}{2})$

$= 4^{8} {\sin \frac{θ}{2} . \cos \frac{θ}{2} . \sin (\frac{π}{8} + \frac{θ}{2}) \cos (\frac{π}{8} + \frac{θ}{2}) . \sin (\frac{π}{4} + \frac{θ}{2}) \cos (\frac{π}{4} + \frac{θ}{2}) \sin (\frac{3 π}{8} + \frac{θ}{2}) \cos (\frac{3 π}{8} + \frac{θ}{2})}$

$= 4^{8} {\frac{\sin θ \sin (\frac{π}{4} + θ) \sin (\frac{π}{4} + θ) \sin (\frac{3 π}{4} + θ)}{2^{4}}}$

$= 4^{6} {\sin θ \cos θ \sin (\frac{π}{4} + θ) \cos (\frac{π}{4} + θ)}$

$= 4^{6} {\frac{\sin 2 θ \sin (\frac{π}{2} + 2 θ)}{4}}$

$= 4^{5} \frac{\sin (4 θ)}{2} = 2^{9} \sin 4 θ$

$P is maximum when \sin 4 θ = 1 \Rightarrow θ = \frac{π}{8}$

$P_{\max} = 2^{9} = 512$

Q 15 :

Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point A = (4, 1), where $1 < r < 3$ . Two distinct common tangents PQ and ST of $C_{1}$ and $C_{2}$ are drawn. The tangent PQ touches $C_{1}$ at P and $C_{2}$ at Q. The tangent ST touches $C_{1}$ at S and $C_{2}$ at T. Mid points of the line segments PQ and ST are joined to form a line which meets the $x$ -axis at a point B. If $A B = \sqrt{5}$ , then the value of $r^{2}$ is [2023]

(2)

[IMAGE 312]

$C_{1} : x^{2} + y^{2} = 1 . . . . (i)$

$Let C_{2} : {(x - 4)}^{2} + {(y - 1)}^{2} = r^{2} . . . (ii)$

$Radical axis:$ $8 x + 2 y - 17 = 1 - r^{2} [from (i) and (ii)]$

$\Rightarrow 8 x + 2 y = 18 - r^{2}$

$B (\frac{18 - r^{2}}{8}, 0) and A (4, 1)$

$Given A B = \sqrt{5}$ $\Rightarrow \sqrt{{(\frac{18 - r^{2}}{8} - 4)}^{2} + 1} = \sqrt{5}$ $\Rightarrow r^{2} = 2$

Q 16 :

Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2}$ , where $r > \frac{\sqrt{5}}{2}$ . Suppose $P Q$ is a chord of this circle and the equation of the line passing through P and Q is $2 x + 4 y = 5$ . If the centre of the circumcircle of the triangle OPQ lies on the line $x + 2 y = 4$ , then the value of $r$ is _____ . [2020]

(2)

[IMAGE 313]

$∵$ $Centre of circle is O (0, 0)$

$O A = perpendicular distance from point O to line$

$2 x + 4 y = 5$ $=$ $| \frac{0 + 0 - 5}{\sqrt{4 + 16}} |$ $= \frac{\sqrt{5}}{2}$

$O C = perpendicular distance from point O to line$

$x + 2 y = 4$ $=$ $| \frac{0 + 0 - 4}{\sqrt{1 + 4}} |$ $= \frac{4}{\sqrt{5}}$

$∴$ $C A = O C - O A = \frac{3}{2 \sqrt{5}}$ $∵$ $C Q = O C = \frac{4}{\sqrt{5}} (radius)$

$Now A Q^{2} = C Q^{2} - C A^{2} (∵ A C ⊥ P Q) = \frac{16}{5} - \frac{9}{20} = \frac{11}{4}$

$∴$ $O Q = r = \sqrt{O A^{2} + A Q^{2}}$

$\Rightarrow r = \sqrt{\frac{5}{4} + \frac{11}{4}} = \sqrt{4} = 2$

Q 17 :

Let the point B be the reflection of the point A(2, 3) with respect to the line $8 x - 6 y - 23 = 0$ . Let ${I^{-}}_{A}$ and ${I^{-}}_{B}$ be circles of radii 2 and 1 with centers A and B respectively. Let T be a common tangent to the circles ${I^{-}}_{A}$ and ${I^{-}}_{B}$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is _______ . [2019]

(10)

$A N =$ $| \frac{16 - 18 - 23}{\sqrt{64 + 36}} |$ $= \frac{25}{10} = \frac{5}{2} = B N$

[IMAGE 314]

$∵$ $∆ C P A ~ ∆ C Q B (By AA similarity)$

$∴$ $\frac{C A}{C B} = \frac{P A}{Q B} \Rightarrow \frac{C A}{C A - 5} = \frac{2}{1}$

$\Rightarrow C A = 2 C A - 10 \Rightarrow C A = 10$

Q 18 :

For how many values of $p$ , the circle $x^{2} + y^{2} + 2 x + 4 y - p = 0$ and the coordinate axes have exactly three common points? [2017]

(2)

$Centre of the circle is (- 1, - 2)$

Geometrically, circle will have exactly 3 common points with axes in the cases

$(i) Passing through origin \Rightarrow p = 0$

$(ii) Touching x -axis and intersecting y -axis at two points$ $i.e. f^{2} > c and g^{2} = c$

$i.e. 4 > - p and 1 = - p \Rightarrow p > - 4 and p = - 1$ $∴$ $p = - 1$

$(iii) Touching y -axis and intersecting x -axis at two points$ $i.e. f^{2} = c and g^{2} > c$

$\Rightarrow 4 = - p and 1 > - p$

$\Rightarrow p = - 4 and p > - 1, which is not possible$

$∴$ $Only two values of p are possible.$

Q 19 :

The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts.

If $S = {(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})}$ , then the number of points(s) in S lying inside the smaller part is [2011]

(2)

The smaller region of circle is the region given by

$x^{2} + y^{2} \leq 6 ...(i)$

and $2 x - 3 y \geq 1 ...(ii)$

[IMAGE 315]

We observe that only two points $(2, \frac{3}{4})$ and $(\frac{1}{4}, - \frac{1}{4})$ satisfy both the inequalities (i) and (ii).

$∴$ $2 points in S lie inside the smaller part.$

Q 20 :

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through P is also a common tangent to $C_{2}$ and $C$ , then the radius of the circle $C$ is [2009]

(8)

$Let r be the radius of required circle.$

$Clearly, in ∆ C_{1} C C_{2}, C_{1} C = C_{2} C = r + 1 and P is mid point of C_{1} C_{2}$

$∴$ $C P ⊥ C_{1} C_{2}, Also P M ⊥ C C_{1}$

[IMAGE 316]

$Now ∆ P M C_{1} ~ ∆ C P C_{1} (By AA similarity)$

$∴$ $\frac{M C_{1}}{P C_{1}} = \frac{P C_{1}}{C C_{1}} \Rightarrow \frac{1}{3} = \frac{3}{r + 1}$ $\Rightarrow r + 1 = 9 \Rightarrow r = 8$

Q 21 :

Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]

(0.84)

$We have A B = 1, A C = 3 and ∠ B A C = \frac{π}{2}$

$Let A be the origin, B on x -axis, C on y -axis as shown below$

[IMAGE 317]

$∴$ $Equation of circumcircle is$

${(x - \frac{1}{2})}^{2} + {(y - \frac{3}{2})}^{2} = {(\sqrt{{(1 - 0)}^{2} + {(0 - 3)}^{2}} \div 2)}^{2} = \frac{5}{2} ...(i)$

$Required circle touches A B and A C, have radius r$

$∴$ $Equation be {(x - r)}^{2} + {(y - r)}^{2} = r^{2} ...(ii)$

$If circle in equation (ii) touches circumcircle internally, we have$ $d_{C_{1} C_{2}} =$ $| r_{1} - r_{2} |$

$\Rightarrow {(\frac{1}{2} - r)}^{2} + {(\frac{3}{2} - r)}^{2} = {(| \sqrt{\frac{5}{2}} - r |)}^{2}$

$\Rightarrow \frac{1}{4} + r^{2} - r + \frac{9}{4} + r^{2} - 3 r$

$\Rightarrow {(\sqrt{\frac{5}{2}} - r)}^{2} or {(r - \sqrt{\frac{5}{2}})}^{2}$

$\Rightarrow 2 r^{2} - 4 r + \frac{5}{2} = \frac{5}{2} + r^{2} - \sqrt{10} r$

$\Rightarrow r = 0 or 4 - \sqrt{10}$

$\Rightarrow r \approx 0.837 \approx 0.84$ (on rounding off)

Q 22 :

Let $G$ be a circle of radius $R > 0$ . Let $G_{1}, G_{2}, \dots, G_{n}$ be $n$ circles of equal radius $r > 0$ . Suppose each of the $n$ circles $G_{1}, G_{2}, \dots, G_{n}$ touches the circle $G$ externally. Also, for $i = 1, 2, \dots, n - 1$ , the circle $G_{i}$ touches $G_{i + 1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE? [2022]

If $n = 4$ , then $(\sqrt{2} - 1) r < R$
If $n = 5$ , then $r < R$
If $n = 8$ , then $(\sqrt{2} - 1) r < R$
If $n = 12$ , then $\sqrt{2} (\sqrt{3} + 1) r > R$

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4

Select one or more options

(3, 4)

[IMAGE 318]

Refer to diagram,

In $∆ A O B$

$\sin (\frac{π}{n}) = \frac{r}{R + r}$

$\Rightarrow c o s e c (\frac{π}{n}) = \frac{R}{r} + 1$

$\Rightarrow R = r [c o s e c (\frac{π}{n}) - 1]$

$If n = 4, R = r (\sqrt{2} - 1)$

$If n = 5, R = r (c o s e c \frac{π}{5} - 1)$

$∵$ $c o s e c \frac{π}{5} < c o s e c \frac{π}{6}$

$(c o s e c \frac{π}{5} - 1) < 2 - 1 = 1$ $∴$ $R < r$

$If n = 8, R = r (c o s e c \frac{π}{8} - 1)$ $∴$ $c o s e c \frac{π}{8} > c o s e c \frac{π}{4}$

$(c o s e c \frac{π}{8} - 1) > \sqrt{2} - 1 \Rightarrow R > r (\sqrt{2} - 1)$

$If n = 12, R = r (c o s e c \frac{π}{12} - 1)$

$R = r (\sqrt{2} (\sqrt{3} + 1) - 1), R < \sqrt{2} (\sqrt{3} + 1) r$

Q 23 :

For any complex number $w = c + i d$ , let $\arg (w) \in (- π, π]$ , where $i = \sqrt{- 1}$ . Let $α$ and $β$ be real numbers such that for all complex numbers $z = x + i y$ satisfying $\arg (\frac{z + α}{z + β}) = \frac{π}{4},$ the ordered pair $(x, y)$ lies on the circle $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$ . Then which of the following statements is (are) TRUE? [2021]

$α = - 1$
$α β = 4$
$α β = - 4$
$β = 4$

1

2

3

4

Select one or more options

(2, 4)

Given that $\arg (\frac{z + α}{z + β}) = \arg (z + α) - \arg (z + β) = \frac{π}{4}$ implies $z$ is on arc and $(- α, 0)$ & $(- β, 0)$ subtend $\frac{π}{4}$ on $z$ .

Given that $z$ lies on $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$

So put $y = 0$ ; for value of $α$ and $β$

$x^{2} + 5 x + 4 = 0 \Rightarrow x = - 1, x = - 4$

[IMAGE 319]

$∴$ $α = 1, β = 4 .$

Q 24 :

A circle $S$ passes through the point (0, 1) and is orthogonal to the circles ${(x - 1)}^{2} + y^{2} = 16$ and $x^{2} + y^{2} = 1$ . Then [2014]

radius of $S$ is 8
radius of $S$ is 7
centre of $S$ is $(- 7, 1)$
centre of $S$ is $(- 8, 1)$

1

2

3

4

Select one or more options

(2, 3)

Let the equation of circles be

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0 ...(i)$

It passes through (0, 1)

$∴$ $1 + 2 f + c = 0 ...(ii)$

Since equation (i) is orthogonal to circle ${(x - 1)}^{2} + y^{2} = 16$

$i.e. x^{2} + y^{2} - 2 x - 15 = 0$

and $x^{2} + y^{2} - 1 = 0$

$∴$ $2 g (- 1) + 2 f (0) = c - 15$

$\Rightarrow 2 g + c - 15 = 0 ...(iii)$

$and 2 g \cdot 0 + 2 f \cdot 0 = c - 1 \Rightarrow c = 1 ...(iv)$

$Solving (ii), (iii) and (iv), we get$

$c = 1, g = 7, f = - 1$

$∴$ $Required circle is x^{2} + y^{2} + 14 x - 2 y + 1 = 0,$ $with centre (- 7, 1) and radius 7$

$∴$ $(2) and (3) are correct options.$

Q 25 :

Circle(s) touching $x$ -axis at a distance 3 from the origin and having an intercept of length $2 \sqrt{7}$ on $y$ -axis is (are) [2013]

$x^{2} + y^{2} - 6 x + 8 y + 9 = 0$
$x^{2} + y^{2} - 6 x + 7 y + 9 = 0$
$x^{2} + y^{2} - 6 x - 8 y + 9 = 0$
$x^{2} + y^{2} - 6 x - 7 y + 9 = 0$

1

2

3

4

Select one or more options

(1, 3)

[IMAGE 320]

Here, there are two possibilities for the given circle as shown in the figure.

$∴$ $The equations of circles can be$

${(x - 3)}^{2} + {(y - 4)}^{2} = 4^{2} or {(x - 3)}^{2} + {(y + 4)}^{2} = 4^{2}$

$\Rightarrow x^{2} + y^{2} - 6 x - 8 y + 9 = 0 or x^{2} + y^{2} - 6 x + 8 y + 9 = 0$

Q 26 :

Let RS be the diameter of the circle $x^{2} + y^{2} = 1$ , where $S$ is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s). [2016]

$(\frac{1}{3}, \frac{1}{\sqrt{3}})$
$(\frac{1}{4}, \frac{1}{2})$
$(\frac{1}{3}, - \frac{1}{\sqrt{3}})$
$(\frac{1}{4}, - \frac{1}{2})$

1

2

3

4

Select one or more options

(1, 3)

Given: A circle : $x^{2} + y^{2} = 1$

Let coordinates of $P = (\cos θ, \sin θ)$

$∴$ $Equation of tangent at P (\cos θ, \sin θ) is$

$x \cos θ + y \sin θ = 1 ...(i)$

$Equation of normal at P is$ $y = x \tan θ ...(ii)$

$Now, equation of tangent at S is x = 1 ...(iii)$

$On solving (i) and (iii), we get the coordinates of Q as$

$(1, \frac{1 - \cos θ}{\sin θ}) = (1, \tan \frac{θ}{2})$

[IMAGE 321]

$∴$ $Equation of line through Q and parallel to R S is$ $y = \tan \frac{θ}{2} ...(iv)$

$Intersection point E of normal (ii) and line (iv) can be find out by solving (ii) and (iv).$

$Now from (ii) and (iv),$

$\tan \frac{θ}{2} = x \tan θ \Rightarrow x = \frac{1 - \tan^{2} \frac{θ}{2}}{2}$

$∴$ $Locus of E is$ $x = \frac{1 - y^{2}}{2} \Rightarrow y^{2} = 1 - 2 x$

$It is satisfied by the points$ $(\frac{1}{3}, \frac{1}{\sqrt{3}}) and (\frac{1}{3}, - \frac{1}{\sqrt{3}})$

Q 27 :

Let the straight line $y = 2 x$ touch a circle with center $(0, α)$ , $α > 0$ , and radius $r$ at a point $A_{1}$ . Let $B_{1}$ be the point on the circle such that the line segment $A_{1} B_{1}$ is a diameter of the circle. Let $α + r = 5 + \sqrt{5}$ .

Match each entry in List-I to the correct entry in List-II.

List-I List-II

(P) $α$ equals (1) $(- 2, 4)$

(Q) $r$ equals (2) $\sqrt{5}$

(R) $A_{1}$ equals (3) $(- 2, 6)$

(S) $B_{1}$ equals (4) $5$

(5) $(2, 4)$

The correct option is [2024]

(P) → (4), (Q) → (2), (R) → (1), (S) → (3)
(P) → (2), (Q) → (4), (R) → (1), (S) → (3)
(P) → (4), (Q) → (2), (R) → (5), (S) → (3)
(P) → (2), (Q) → (4), (R) → (3), (S) → (5)

1

2

3

4

(3)

[IMAGE 322]

$Consider centre as P (0, α), α > 0$

$Distance of A_{1} P =$ $| \frac{2 (0) - α}{\sqrt{5}} |$ $= r$

$\Rightarrow$ $| - α |$ $= \sqrt{5} r \Rightarrow α = \sqrt{5} r$ $∴$ $α + r = 5 + \sqrt{5}$

$\Rightarrow \sqrt{5} r + r = \sqrt{5} (\sqrt{5} + 1) \Rightarrow r = \sqrt{5}, α = 5$

$∴$ $P (0, 5)$

$Foot of perpendicular from P to line 2 x - y = 0$

$\frac{x - 0}{2} = \frac{y - 5}{- 1} = \frac{- (2 (0) - 5)}{5} = 1$

$\Rightarrow x = 2, y = 4 A_{1} (2, 4)$

$Let B (x_{1}, y_{1}),$ $∴$ $\frac{x_{1} + 2}{2} = 0, \frac{y_{1} + 4}{2} = 5$

$\Rightarrow x_{1} = - 2, y_{1} = 6 B (- 2, 6)$

Q 28 :

Let the circles $C_{1} : x^{2} + y^{2} = 9$ and $C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16$ , intersect at the points $X$ and $Y$ . Suppose that another circle $C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ satisfies the following conditions:

(i) Centre of $C_{3}$ is collinear with the centres of $C_{1}$ and $C_{2}$ ,

(ii) $C_{1}$ and $C_{2}$ both lie inside $C_{3}$ , and

(iii) $C_{3}$ touches $C_{1}$ at $M$ and $C_{2}$ at $N$ .

Let the line through $X$ and $Y$ intersect $C_{3}$ at $Z$ and $W$ , and let a common tangent of $C_{1}$ and $C_{3}$ be a tangent to the parabola $x^{2} = 8 α y$ .

There are some expressions given in the Column-I whose values are given in Column-II below:

Column I Column II

(A) $2 h + k$ (p) $6$

(B) $\frac{Length of Z W}{Length of X Y}$ (q) $\sqrt{6}$

(C) $\frac{Area of triangle M Z N}{Area of triangle Z M W}$ (r) $\frac{5}{4}$

(D) $α$ (s) $\frac{21}{5}$

(t) $2 \sqrt{6}$

(u) $\frac{10}{3}$

Q. Which of the following is the only CORRECT combination? [2019]

$(A), (u)$
$(A), (s)$
$(B), (t)$
$(B), (q)$

1

2

3

4

(4)

[IMAGE 323]

$Given three circles are$

$C_{1} : x^{2} + y^{2} = 9$

$C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16$

$C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

$Centres of circles C_{1}, C_{2}, C_{3} are D (0, 0), E (3, 4), F (h, k) respectively$

$and radii of circles C_{1} : C_{2} : C_{3} are 3, 4, r respectively$

$Equation of D E : y = \frac{4}{3} x$

$Centres of circles C_{1}, C_{2}, C_{3} are collinear \Rightarrow F (h, \frac{4}{3} h)$

$M N = M D + D E + E N = 3 + 5 + 4 = 12 \Rightarrow r = 6$

$∴$ $D E = 6 - 3 = 3$

$\Rightarrow h^{2} + \frac{16}{9} h^{2} = 9 \Rightarrow h^{2} = \frac{81}{25} \Rightarrow h = \frac{9}{5}$ $taking h +ve, as lies between D and E$

$∴$ $F (\frac{9}{5}, \frac{12}{5})$

$∴$ $2 h + k = \frac{18}{5} + \frac{12}{5} = \frac{30}{5} = 6$

$DE is common chord of circles C_{1} and C_{2}$

$∴$ $Equation of X Y : S_{1} - S_{2} = 0$

$\Rightarrow 6 x + 8 y - 18 = 0 \Rightarrow 3 x + 4 y - 9 = 0$

$Length of perpendicular from D to X Y = \frac{9}{5} = D P$

$Also D X = 3, P X = \sqrt{9 - \frac{81}{25}} = \sqrt{\frac{225 - 81}{25}} = \frac{12}{5}$

$∴$ $X Y = 2 P X = \frac{24}{5}$

$ZW is chord of C_{3}$

$F P = M F - M P = 6 - (3 + \frac{9}{5}) = 6 - \frac{24}{5} = \frac{6}{5}$

$∴$ $Z P = \sqrt{6^{2} - {(\frac{6}{5})}^{2}} = \frac{12 \sqrt{6}}{5}$ $∴$ $Z W = \frac{24 \sqrt{6}}{5}$

$Hence, \frac{Length of Z W}{Length of X Y} = \frac{24 \sqrt{6} / 5}{24 / 5} = \sqrt{6}$

$∴$ $(B) - (q)$

$Area of ∆ M Z N = \frac{1}{2} \times M N \times Z P = \frac{1}{2} \times 12 \times \frac{12 \sqrt{6}}{5} = \frac{72 \sqrt{6}}{5}$

$Area of ∆ Z M W = \frac{1}{2} \times Z W \times M P = \frac{1}{2} \times \frac{24 \sqrt{6}}{5} \times \frac{24}{5} = \frac{288 \sqrt{6}}{25}$

$∴$ $\frac{Area of ∆ M Z N}{Area of ∆ Z M W} = \frac{72 \sqrt{6}}{5} \times \frac{25}{288 \sqrt{6}} = \frac{5}{4}$

$∴$ $(C) - (r)$

$Now common tangent of C_{1} and C_{3} is S_{1} - S_{3} = 0$

$\Rightarrow 2 h x + 2 k y - h^{2} - k^{2} = 9 - r^{2}$

$\Rightarrow \frac{18}{5} x + \frac{24}{5} y - \frac{81}{25} - \frac{144}{25} = 9 - 36$

$\Rightarrow 3 x + 4 y + 15 = 0$

$It is tangent to x^{2} = 8 α y$

Putting value of $y$ from common tangent in parabola, we get

$\Rightarrow x^{2} = - 8 α (\frac{3 x + 15}{4})$ $\Rightarrow x^{2} + 6 α x + 30 α = 0$

It should have equal roots

$∴$ $36 α^{2} - 120 α = 0 \Rightarrow α = \frac{10}{3}$

$∴$ $(D) - (u)$

Thus $(B) - (q)$ is the only correct combination and $(D) - (s)$ is the only incorrect combination.

Option (4) is correct

Q 29 :

Let the circles $C_{1} : x^{2} + y^{2} = 9$ and $C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16$ , intersect at the points $X$ and $Y$ . Suppose that another circle $C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = p^{2}$ satisfies the following conditions:

(i) Centre of $C_{3}$ is collinear with the centres of $C_{1}$ and $C_{2}$

(ii) $C_{1}$ and $C_{2}$ both lie inside $C_{3}$ , and

(iii) $C_{3}$ touches $C_{1}$ at $M$ and $C_{2}$ at $N$ .

Let the line through $X$ and $Y$ intersect $C_{3}$ at $Z$ and $W$ , and let a common tangent of $C_{1}$ and $C_{3}$ be a tangent to the parabola $x^{2} = 8 α y$ .

There are some expressions given in the Column-I whose values are given in Column-II below:

Column I Column II

(A) $2 h + k$ (p) $6$

(B) $\frac{Length of Z W}{Length of X Y}$ (q) $\sqrt{6}$

(C) $\frac{Area of triangle M Z N}{Area of triangle Z M W}$ (r) $\frac{5}{4}$

(D) $α$ (s) $\frac{21}{5}$

(t) $2 \sqrt{6}$

(u) $\frac{10}{3}$

Q. Which of the following is the only INCORRECT combination? [2019]

$(D), (s)$
$(A), (p)$
$(C), (r)$
$(D), (u)$

1

2

3

4

(1)

[IMAGE 324]

$Given three circles are$

$C_{1} : x^{2} + y^{2} = 9$

$C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16$

$C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

$Centres of circles C_{1}, C_{2}, C_{3} are D (0, 0), E (3, 4), F (h, k) respectively$

$and radii of circles C_{1} : C_{2} : C_{3} are 3, 4, r respectively$

$Equation of D E : y = \frac{4}{3} x$

$Centres of circles C_{1}, C_{2}, C_{3} are collinear \Rightarrow F (h, \frac{4}{3} h)$

$M N = M D + D E + E N = 3 + 5 + 4 = 12 \Rightarrow r = 6$

$∴$ $D E = 6 - 3 = 3$

$\Rightarrow h^{2} + \frac{16}{9} h^{2} = 9 \Rightarrow h^{2} = \frac{81}{25} \Rightarrow h = \frac{9}{5}$ $taking h +ve, as lies between D and E$

$∴$ $F (\frac{9}{5}, \frac{12}{5})$

$\Rightarrow 2 h + k = \frac{18}{5} + \frac{12}{5} = \frac{30}{5} = 6$

$∴$ $(A) - (p)$

$DE is common chord of circles C_{1} and C_{2}$

$∴$ $Equation of X Y : S_{1} - S_{2} = 0$

$\Rightarrow 6 x + 8 y - 18 = 0 \Rightarrow 3 x + 4 y - 9 = 0$

$Length of perpendicular from D to X Y = \frac{9}{5} = D P$

$Also D X = 3, P X = \sqrt{9 - \frac{81}{25}} = \sqrt{\frac{225 - 81}{25}} = \frac{12}{5}$

$∴$ $X Y = 2 P X = \frac{24}{5}$

$ZW is chord of C_{3}$

$F P = M F - M P = 6 - (3 + \frac{9}{5}) = 6 - \frac{24}{5} = \frac{6}{5}$

$∴$ $Z P = \sqrt{6^{2} - {(\frac{6}{5})}^{2}} = \frac{12 \sqrt{6}}{5}$ $∴$ $Z W = \frac{24 \sqrt{6}}{5}$

$Hence, \frac{Length of Z W}{Length of X Y} = \frac{24 \sqrt{6} / 5}{24 / 5} = \sqrt{6}$

$∴$ $(B) - (q)$

$Area of ∆ M Z N = \frac{1}{2} \times M N \times Z P = \frac{1}{2} \times 12 \times \frac{12 \sqrt{6}}{5} = \frac{72 \sqrt{6}}{5}$

$Area of ∆ Z M W = \frac{1}{2} \times Z W \times M P = \frac{1}{2} \times \frac{24 \sqrt{6}}{5} \times \frac{24}{5} = \frac{288 \sqrt{6}}{25}$

$∴$ $\frac{Area of ∆ M Z N}{Area of ∆ Z M W} = \frac{72 \sqrt{6}}{5} \times \frac{25}{288 \sqrt{6}} = \frac{5}{4}$

$∴$ $(C) - (r)$

$Now common tangent of C_{1} and C_{3} is S_{1} - S_{3} = 0$

$\Rightarrow 2 h x + 2 k y - h^{2} - k^{2} = 9 - r^{2}$

$\Rightarrow \frac{18}{5} x + \frac{24}{5} y - \frac{81}{25} - \frac{144}{25} = 9 - 36$

$\Rightarrow 3 x + 4 y + 15 = 0$

$It is tangent to x^{2} = 8 α y$

Putting value of $y$ from common tangent in parabola, we get

$\Rightarrow x^{2} = - 8 α (\frac{3 x + 15}{4})$ $\Rightarrow x^{2} + 6 α x + 30 α = 0$

It should have equal roots

$∴$ $36 α^{2} - 120 α = 0 \Rightarrow α = \frac{10}{3}$

$∴$ $(D) - (u)$

Thus $(B) - (q)$ is the only correct combination and $(D) - (s)$ is the only incorrect combination.

Option (1) is incorrect

Q 30 :

Let $M = {(x, y) \in ℝ \times ℝ : x^{2} + y^{2} \leq r^{2}}$ where $r > 0$ .

Consider the geometric progression $a_{n} = \frac{1}{2^{n - 1}}, n = 1, 2, 3, \dots$ . Let $S_{0} = 0$ and, for $n \geq 1$ , let $S_{n}$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$ , let $C_{n}$ denote the circle with center $(S_{n - 1}, 0)$ and radius $a_{n}$ and $D_{n}$ denote the circle with center $(S_{n - 1}, S_{n - 1})$ and radius $a_{n}$ . [2021]

Q . Consider $M$ with $r = \frac{1025}{513}$ . Let $k$ be the number of all those circles $C_{n}$ that are inside $M$ . Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

$k + 2 l = 22$
$2 k + l = 26$
$2 k + 3 l = 34$
$3 k + 2 l = 40$

1

2

3

4

(4)

$∵$ $a_{n} = \frac{1}{2^{n - 1}}$

$S_{n} = 1 + \frac{1}{2} + \frac{1}{2^{2}} + \dots + \frac{1}{2^{n - 1}} = 2 (1 - \frac{1}{2^{n}}) = 2 - \frac{1}{2^{n - 1}}$

$For circles C_{n} to lie inside M$

$S_{n - 1} + a_{n} < \frac{1025}{513} \Rightarrow 2 - \frac{1}{2^{n - 2}} + \frac{1}{2^{n - 1}} < \frac{1025}{513}$

$\Rightarrow 1 - \frac{1}{2^{n}} < \frac{1025}{1026} = 1 - \frac{1}{1026}$

$\Rightarrow 2^{n} < 1026 \Rightarrow n \leq 10 \Rightarrow k = 10$

$Also l = 5$

$3 k + 2 l = 30 + 10 = 40$

Topic Question Set

Q 1 :

Consider a triangle $Δ$ whose two sides lie on the $x$ -axis and the line $x + y + 1 = 0$ . If the orthocenter of $Δ$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $Δ$ is [2021]

Q 2 :

A line $y = m x + 1$ intersects the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ at the points P and Q. If the midpoint of the line segment PQ has $x$ -coordinate $- \frac{3}{5}$ , then which one of the following options is correct? [2019]

Q 3 :

The circle passing through the point $(- 1, 0)$ and touching the $y$ -axis at (0, 2) also passes through the point. [2011]

Q 4 :

Tangents drawn from the point P(1, 8) to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the points A and B. The equation of the circumcircle of the triangle $P A B$ is [2009]

Q 5 :

If one of the diameters of the circle $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ is a chord to the circle with centre (2, 1), then the radius of the circle is [2004]

Q 6 :

The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]

Q 7 :

A circle is given by $x^{2} + {(y - 1)}^{2} = 1$ , another circle $C$ touches it externally and also the $x$ -axis, then the locus of its centre is [2005]

Q 8 :

The centre of circle inscribed in square formed by the lines $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ , is [2003]

Q 9 :

If the tangent at the point $P$ on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets a straight line $5 x - 2 y + 6 = 0$ at a point $Q$ on the $y$ -axis, then the length of $P Q$ is [2002]

Q 10 :

Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$ . If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals [2001]

Q 11 :

Let $A B$ be a chord of the circle $x^{2} + y^{2} = r^{2}$ subtending a right angle at the centre. Then the locus of the centroid of the triangle $P A B$ as $P$ moves on the circle is [2001]

Q 12 :

If the circles $x^{2} + y^{2} + 2 x + 2 k y + 6 = 0$ , $x^{2} + y^{2} + 2 k y + k = 0$ intersect orthogonally, then $k$ is [2000]

Q 13 :

The triangle $P Q R$ is inscribed in the circle $x^{2} + y^{2} = 25$ . If $Q$ and $R$ have co-ordinates (3, 4) and $(- 4, 3)$ respectively, then $∠ Q P R$ is equal to [2000]

Q 18 :

For how many values of $p$ , the circle $x^{2} + y^{2} + 2 x + 4 y - p = 0$ and the coordinate axes have exactly three common points? [2017]

Q 19 :

The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts.

If $S = {(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})}$ , then the number of points(s) in S lying inside the smaller part is [2011]

Q 21 :

Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]

Q 24 :

A circle $S$ passes through the point (0, 1) and is orthogonal to the circles ${(x - 1)}^{2} + y^{2} = 16$ and $x^{2} + y^{2} = 1$ . Then [2014]

Q 25 :

Circle(s) touching $x$ -axis at a distance 3 from the origin and having an intercept of length $2 \sqrt{7}$ on $y$ -axis is (are) [2013]

Want Us to Email you About Special Offers & Updates?

	List-I		List-II
(P)	$α$ equals	(1)	$(- 2, 4)$
(Q)	$r$ equals	(2)	$\sqrt{5}$
(R)	$A_{1}$ equals	(3)	$(- 2, 6)$
(S)	$B_{1}$ equals	(4)	$5$
		(5)	$(2, 4)$

	Column I		Column II
(A)	$2 h + k$	(p)	$6$
(B)	$\frac{Length of Z W}{Length of X Y}$	(q)	$\sqrt{6}$
(C)	$\frac{Area of triangle M Z N}{Area of triangle Z M W}$	(r)	$\frac{5}{4}$
(D)	$α$	(s)	$\frac{21}{5}$
		(t)	$2 \sqrt{6}$
		(u)	$\frac{10}{3}$

Topic Question Set

Q 1 : Consider a triangle Δ whose two sides lie on the x-axis and the line x+y+1=0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is [2021]

Q 2 : A line y=mx+1 intersects the circle (x-3)2+(y+2)2=25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate -35, then which one of the following options is correct? [2019]

Q 3 : The circle passing through the point (-1,0) and touching the y-axis at (0, 2) also passes through the point. [2011]

Q 4 : Tangents drawn from the point P(1, 8) to the circle x2+y2-6x-4y-11=0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is [2009]

Q 5 : If one of the diameters of the circle x2+y2-2x-6y+6=0 is a chord to the circle with centre (2, 1), then the radius of the circle is [2004]

Q 6 : The common tangents to the circle x2+y2=2 and the parabola y2=8x touch the circle at the points P,Q and the parabola at the points R,S. Then the area of the quadrilateral PQRS is [2014]

Q 7 : A circle is given by x2+(y-1)2=1, another circle C touches it externally and also the x-axis, then the locus of its centre is [2005]

Q 8 : The centre of circle inscribed in square formed by the lines x2-8x+12=0 and y2-14y+45=0, is [2003]

Q 9 : If the tangent at the point P on the circle x2+y2+6x+6y=2 meets a straight line 5x-2y+6=0 at a point Q on the y-axis, then the length of PQ is [2002]

Q 10 : Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals [2001]

Q 11 : Let AB be a chord of the circle x2+y2=r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB as P moves on the circle is [2001]

Q 12 : If the circles x2+y2+2x+2ky+6=0, x2+y2+2ky+k=0 intersect orthogonally, then k is [2000]

Q 13 : The triangle PQR is inscribed in the circle x2+y2=25. If Q and R have co-ordinates (3, 4) and (-4,3) respectively, then ∠QPR is equal to [2000]

Q 14 : Let A1,A2,A3,…,A8 be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let PAi denote the distance between the points P and Ai for i=1,2,…,8. If P varies over the circle, then the maximum value of the product PA1·PA2⋯PA8 is [2023]

Q 16 : Let O be the centre of the circle x2+y2=r2, where r>52. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x+4y=5. If the centre of the circumcircle of the triangle OPQ lies on the line x+2y=4, then the value of r is _____ . [2020]

Q 18 : For how many values of p, the circle x2+y2+2x+4y-p=0 and the coordinate axes have exactly three common points? [2017]

Q 19 : The straight line 2x-3y=1 divides the circular region x2+y2≤6 into two parts. If S={(2,34),(52,34),(14,-14),(18,14)}, then the number of points(s) in S lying inside the smaller part is [2011]

Q 21 : Let ABC be the triangle with AB=1, AC=3 and ∠BAC=π2. If a circle of radius r>0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is _______. [2022]

Q 23 : For any complex number w=c+id, let arg(w)∈(-π,π], where i=-1. Let α and β be real numbers such that for all complex numbers z=x+iy satisfying arg(z+αz+β)=π4, the ordered pair (x,y) lies on the circle x2+y2+5x-3y+4=0. Then which of the following statements is (are) TRUE? [2021]

Q 24 : A circle S passes through the point (0, 1) and is orthogonal to the circles (x-1)2+y2=16 and x2+y2=1. Then [2014]

Q 25 : Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 27 on y-axis is (are) [2013]

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Q 1 :

Consider a triangle $Δ$ whose two sides lie on the $x$ -axis and the line $x + y + 1 = 0$ . If the orthocenter of $Δ$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $Δ$ is [2021]

Q 2 :

A line $y = m x + 1$ intersects the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ at the points P and Q. If the midpoint of the line segment PQ has $x$ -coordinate $- \frac{3}{5}$ , then which one of the following options is correct? [2019]

Q 3 :

The circle passing through the point $(- 1, 0)$ and touching the $y$ -axis at (0, 2) also passes through the point. [2011]

Q 4 :

Tangents drawn from the point P(1, 8) to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the points A and B. The equation of the circumcircle of the triangle $P A B$ is [2009]

Q 5 :

If one of the diameters of the circle $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ is a chord to the circle with centre (2, 1), then the radius of the circle is [2004]

Q 6 :

The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]

Q 7 :

A circle is given by $x^{2} + {(y - 1)}^{2} = 1$ , another circle $C$ touches it externally and also the $x$ -axis, then the locus of its centre is [2005]

Q 8 :

The centre of circle inscribed in square formed by the lines $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ , is [2003]

Q 9 :

If the tangent at the point $P$ on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets a straight line $5 x - 2 y + 6 = 0$ at a point $Q$ on the $y$ -axis, then the length of $P Q$ is [2002]

Q 10 :

Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$ . If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals [2001]

Q 11 :

Let $A B$ be a chord of the circle $x^{2} + y^{2} = r^{2}$ subtending a right angle at the centre. Then the locus of the centroid of the triangle $P A B$ as $P$ moves on the circle is [2001]

Q 12 :

If the circles $x^{2} + y^{2} + 2 x + 2 k y + 6 = 0$ , $x^{2} + y^{2} + 2 k y + k = 0$ intersect orthogonally, then $k$ is [2000]

Q 13 :

The triangle $P Q R$ is inscribed in the circle $x^{2} + y^{2} = 25$ . If $Q$ and $R$ have co-ordinates (3, 4) and $(- 4, 3)$ respectively, then $∠ Q P R$ is equal to [2000]

Q 18 :

For how many values of $p$ , the circle $x^{2} + y^{2} + 2 x + 4 y - p = 0$ and the coordinate axes have exactly three common points? [2017]

Q 19 :

The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts.

If $S = {(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})}$ , then the number of points(s) in S lying inside the smaller part is [2011]

Q 21 :

Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]

Q 24 :

A circle $S$ passes through the point (0, 1) and is orthogonal to the circles ${(x - 1)}^{2} + y^{2} = 16$ and $x^{2} + y^{2} = 1$ . Then [2014]

Q 25 :

Circle(s) touching $x$ -axis at a distance 3 from the origin and having an intercept of length $2 \sqrt{7}$ on $y$ -axis is (are) [2013]