Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 11 :

A line segment AB of length $λ$ moves such that the points A and B remain on the periphery of a circle of radius $λ$ . Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

$\frac{\sqrt{19}}{5} λ$
$\frac{2}{3} λ$
$\frac{\sqrt{19}}{7} λ$
$\frac{3}{5} λ$

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(1)

$∠ O A P = 60 °$ $[∵ O A = O B = A B = λ]$

$A P = \frac{2 λ}{5} (Given)$

$\cos 60 ° = \frac{O A^{2} + A P^{2} - O P^{2}}{2 \cdot O A \cdot A P}$

$\Rightarrow$ $\frac{1}{2} = \frac{λ^{2} + \frac{4 λ^{2}}{25} - O P^{2}}{2 λ \cdot \frac{2 λ}{5}}$

$\Rightarrow \frac{2 λ^{2}}{5} = \frac{29 λ^{2}}{25} - O P^{2} \Rightarrow O P^{2} = \frac{29}{25} λ^{2} - \frac{2}{5} λ^{2} = \frac{19}{25} λ^{2}$

Required locus is $x^{2} + y^{2} = \frac{19}{25} λ^{2}$

$∴$ $Radius = \frac{\sqrt{19}}{5} λ$

Q 12 :

Let A be the point (1, 2) and B be any point on the curve $x^{2} + y^{2} = 16$ . If the centre of the locus of the point P, which divides the line segment AB in the ratio 3 : 2, is the point $C (α, β)$ , then the length of the line segment AC is [2023]

$\frac{2 \sqrt{5}}{5}$
$\frac{3 \sqrt{5}}{5}$
$\frac{6 \sqrt{5}}{5}$
$\frac{4 \sqrt{5}}{5}$

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(2)

$Since, P (h, k) divides A B in the ratio 3 : 2 .$

So, $\frac{12 \cos θ + 2}{5} = h and \frac{12 \sin θ + 4}{5} = k$

$\Rightarrow 12 \cos θ = 5 h - 2$ ...(i)

$and 12 \sin θ = 5 k - 4$ ...(ii)

Squaring (i) and (ii) and then adding, we get

$144 = {(5 h - 2)}^{2} + {(5 k - 4)}^{2}$

$\Rightarrow {(x - \frac{2}{5})}^{2} + {(y - \frac{4}{5})}^{2} = \frac{144}{25}$

Centre $\equiv (\frac{2}{5}, \frac{4}{5}) \equiv C (α, β)$

$A C = \sqrt{{(1 - \frac{2}{5})}^{2} + {(2 - \frac{4}{5})}^{2}} = \sqrt{\frac{9}{25} + \frac{36}{25}} = \frac{3 \sqrt{5}}{5}$

Q 13 :

Let the centre of a circle C be $(α, β)$ and its radius $r < 8$ . Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to C. Then $(α - β + r)$ is equal to [2023]

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5
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(3)

$After solving 4 x + 3 y = 1 and 3 x - 4 y = 32, Point A is (4, - 5)$

$Centre (α, β) lies on the line 4 x + 3 y = 1$

$So, 4 α + 3 β = 1$

$∴$ $β = \frac{1 - 4 α}{3}$

$Now, distance from Centre to line 3 x - 4 y - 32 = 0$ $and 3 x + 4 y - 24 = 0 are equal.$

$So,$ $| \frac{3 α - 4 (\frac{1 - 4 α}{3}) - 32}{\sqrt{3^{2} + 4^{2}}} |$ $=$ $| \frac{3 α + 4 (\frac{1 - 4 α}{3}) - 24}{5} |$

$\Rightarrow$ $| \frac{3 α - 4 (\frac{1 - 4 α}{3}) - 32}{5} |$ $=$ $| \frac{3 α + 4 (\frac{1 - 4 α}{3}) - 24}{5} |$

$\Rightarrow$ $| \frac{9 α - 4 + 16 α - 96}{15} |$ $=$ $| \frac{9 α + 4 - 16 α - 72}{15} |$

$\Rightarrow 25 α - 100 = \pm (- 7 α - 68)$

$∴$ $α = 1 and α = \frac{28}{3}$

For $α = 1$ , Centre is $(1, - 1)$

and $radius is \sqrt{{(4 - 1)}^{2} + {(- 5 + 1)}^{2}} = 5 units$

For $α = \frac{28}{3}$ , $β = \frac{- 109}{9}, Centre$ $(\frac{28}{3}, \frac{- 109}{9})$

$∴$ $radius≃ 8.88 (rejected)$

Hence, $α = 1, β = - 1, r = 5$

$∴$ $α - β + r = 7$

Q 14 :

The number of common tangents to the circles $x^{2} + y^{2} - 18 x - 15 y + 131 = 0$ and $x^{2} + y^{2} - 6 x - 6 y - 7 = 0$ , is [2023]

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(2)

$C_{1} : x^{2} + y^{2} - 18 x - 15 y + 131 = 0$

$C_{2} : x^{2} + y^{2} - 6 x - 6 y - 7 = 0$

We know that for equation of circle

$x^{2} + y^{2} + 2 g x + 2 f y + C = 0$

Centre is $(- g, - f)$ and radius $= \sqrt{g^{2} + f^{2} - C}$

For $C_{1},$ $Centre (A) = (9, \frac{15}{2})$

$and radius (r_{1}) = \sqrt{81 + \frac{225}{4} - 131} = \sqrt{\frac{25}{4}} = \frac{5}{2}$

For $C_{2},$ $Centre (B) = (3, 3)$

$and radius (r_{2}) = \sqrt{9 + 9 + 7} = 5$

and $A B = \sqrt{{(6)}^{2} + {(\frac{9}{2})}^{2}} = \sqrt{36 + \frac{81}{4}} = \sqrt{\frac{225}{4}} = \frac{15}{2} = r_{1} + r_{2}$

$∴$ $There are 3 common tangents$

Q 15 :

The area enclosed by the closed curve C given by the differential equation $\frac{d y}{d x} + \frac{x + a}{y - 2} = 0, y (1) = 0$ is $4 π$ . Let P and Q be the points of intersection of the curve C and the $y$ -axis. If normals at P and Q on the curve C intersect the $x$ -axis at points R and S respectively, then the length of the line segment RS is [2023]

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

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(1)

$\frac{d y}{d x} + \frac{x + a}{y - 2} = 0, y (1) = 0;$ $\frac{d y}{d x} = \frac{- (x + a)}{y - 2}$

$\Rightarrow \int (y - 2) d y = - \int (x + a) d x \Rightarrow \frac{y^{2}}{2} - 2 y = \frac{- x^{2}}{2} - a x + c$

$\Rightarrow \frac{x^{2} + y^{2}}{2} + a x - 2 y = c \Rightarrow x^{2} + y^{2} + 2 a x - 4 y = c_{1} (∵ c_{1} = 2 c)$

$Substituting x = 1 and y = 0 as y (1) = 0$

$\Rightarrow 1 + 2 a = c_{1} ∴ x^{2} + y^{2} + 2 a x - 4 y = 1 + 2 a$

$This represents the equation of a circle having area 4 π$

$∴$ $Radius of circle = 2$

$Now r^{2} = 2^{2} = {(- a)}^{2} + {(2)}^{2} = 1 + 2 a$

$\Rightarrow 0 = {(a + 1)}^{2} \Rightarrow a = - 1$

$∴$ $Equation of circle is x^{2} + y^{2} - 2 x - 4 y = - 1$

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

$\Rightarrow {(x - 1)}^{2} + {(y - 2)}^{2} = 4$

$Now, substitute x = 0 to get intersection point of circle with y-axis$

$\Rightarrow {(y - 2)}^{2} = 3$

$\Rightarrow y = \pm \sqrt{3} + 2$ $∴$ $P (0, \pm \sqrt{3} + 2)$

$P (0, \pm \sqrt{3} + 2) and Q (0, 2 - \sqrt{3}) are the points.$

$Now normal at P will pass through centre (1, 2)$

$∴$ $Equation of line passing through P and (1, 2) is$

$y - (\sqrt{3} + 2) = \frac{- \sqrt{3}}{1} (x)$

$Now, this line will cut x-axis at R (\frac{\sqrt{3} + 2}{\sqrt{3}}, 0)$

$Also, equation of line passing through Q (0, 2 - \sqrt{3}) and (1, 2) is$ $y - (2 - \sqrt{3}) = \sqrt{3} x$

$This will cut the x-axis at S (\frac{\sqrt{3} - 2}{\sqrt{3}}, 0)$

$R S = \sqrt{{(\frac{\sqrt{3} + 2}{\sqrt{3}} - \frac{\sqrt{3} - 2}{\sqrt{3}})}^{2} + {(0 - 0)}^{2}} = \sqrt{{(\frac{4}{\sqrt{3}})}^{2}} = \frac{4}{\sqrt{3}} = \frac{4 \sqrt{3}}{3}$

Q 16 :

The locus of the mid points of the chords of the circle $C_{1} : {(x - 4)}^{2} + {(y - 5)}^{2} = 4$ which subtend an angle $θ_{i}$ at the centre of the circle $C_{1}$ , is a circle of radius $r_{i}$ . If $θ_{1} = \frac{π}{3}, θ_{3} = \frac{2 π}{3}$ and $r_{1}^{2} = r_{2}^{2} + r_{3}^{2},$ then $θ_{2}$ is equal to [2023]

$\frac{π}{2}$
$\frac{π}{4}$
$\frac{π}{6}$
$\frac{0 - π}{4}$

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(1)

If a chord of circle of radius $R$ subtends angle $θ_{i}$ at the centre $C_{1},$ then locus of the end point of this chord in a circle of radius

$r_{i} = R \cos (\frac{θ_{i}}{2})$

Given, $r_{1}^{2} = r_{2}^{2} + r_{3}^{2}$

$\Rightarrow \cos^{2} \frac{θ_{1}}{2} = \cos^{2} \frac{θ_{2}}{2} + \cos^{2} \frac{θ_{3}}{2}$

$\Rightarrow \cos^{2} \frac{π}{6} = \cos^{2} \frac{θ_{2}}{2} + \cos^{2} \frac{π}{3}$

$\Rightarrow \frac{3}{4} = \cos^{2} \frac{θ_{2}}{2} + \frac{1}{4}$

$\Rightarrow \cos^{2} \frac{θ_{2}}{2} = \frac{1}{2} \Rightarrow \cos \frac{θ_{2}}{2} = \frac{1}{\sqrt{2}} \Rightarrow \frac{θ_{2}}{2} = \frac{π}{4} \Rightarrow θ_{2} = \frac{π}{2}$

Q 17 :

The points of intersection of the line $a x + b y = 0 (a \neq b)$ and the circle $x^{2} + y^{2} - 2 x = 0$ are $A (α, 0)$ and $B (1, β)$ . The image of the circle with $A B$ as a diameter in the line $x + y + 2 = 0$ is [2023]

$x^{2} + y^{2} + 3 x + 3 y + 4 = 0$
$x^{2} + y^{2} + 5 x + 5 y + 12 = 0$
$x^{2} + y^{2} + 3 x + 5 y + 8 = 0$
$x^{2} + y^{2} - 5 x - 5 y + 12 = 0$

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(2)

Clearly the line and circle intersect at (0, 0)

$\Rightarrow α = 0$

$(1, β)$ lies on the circle

$1 + β^{2} - 2 \times 1 = 0 \Rightarrow β^{2} = 1 \Rightarrow β = \pm 1$

Let $β = - 1$

(1, -1) should lie on the line

$\Rightarrow a - b = 0 \Rightarrow a = b but a \neq b$

Therefore $β = 1$ .

The line and circle intersect at (0, 0) and (1, 1).

Centre of circle with diameter AB is $(\frac{1}{2}, \frac{1}{2})$

$Radius of circle = \frac{1}{\sqrt{2}}, where A (0, 0) and B (1, 1)$

Let (p, q) be the mirror image of $(\frac{1}{2}, \frac{1}{2})$ w.r.t. $x + y + 2 = 0$

$\frac{p - 1 / 2}{1} = \frac{q - 1 / 2}{1} = \frac{- 2 (\frac{1}{2} + \frac{1}{2} + 2)}{2} = - 3$ $\Rightarrow p = \frac{- 5}{2}, q = \frac{- 5}{2}$

(p, q) is centre of required circle whose equation is

${(x + \frac{5}{2})}^{2} + {(y + \frac{5}{2})}^{2} = \frac{1}{2} \Rightarrow x^{2} + y^{2} + 5 x + 5 y + \frac{25}{2} - \frac{1}{2} = 0$

$\Rightarrow x^{2} + y^{2} + 5 x + 5 y + 12 = 0$

Q 18 :

Let the tangents at the points $A (4, - 11)$ and $B (8, - 5)$ on the circle $x^{2} + y^{2} - 3 x + 10 y - 15 = 0,$ intersect at the point $C$ . Then the radius of the circle whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to [2023]

$2 \sqrt{13}$
$\frac{2 \sqrt{13}}{3}$
$\sqrt{13}$
$\frac{3 \sqrt{3}}{4}$

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(2)

$Equation of tangent at point A (4, - 11) is$

$4 x - 11 y - \frac{3}{2} (x + 4) + 5 (y - 11) - 15 = 0$

$5 x - 12 y = 152$ ...(i)

$Equation of tangent at point B (8, - 5) is$

$8 x - 5 y - \frac{3}{2} (x + 8) + 5 (y - 5) - 15 = 0$

$\Rightarrow x = 8$ ...(ii)

Solving (i) and (ii) we get

$x = 8 and y = \frac{- 28}{3}$

So, point $C$ is $(8, - \frac{28}{3})$ and this point is centre of circle whose tangent is line $A B$ .

Equation of line $A B$ is, $3 x - 2 y = 34$

Perpendicular distance of point $C$ from line $A B$ gives the radius of another circle whose centre is $C (8, - \frac{28}{3}) .$

So, $\frac{| 3 \times 8 - 2 (\frac{- 28}{3}) - 34 |}{\sqrt{3^{2} + {(- 2)}^{2}}} = \frac{2 \sqrt{13}}{3}$

Q 19 :

Let $y = x + 2, 4 y = 3 x + 6$ and $3 y = 4 x + 1$ be three tangent lines to the circle ${(x - h)}^{2} + {(y - k)}^{2} = r^{2} .$ Then $h + k$ is equal to [2023]

5
$5 (1 + \sqrt{2})$
6
$5 \sqrt{2}$

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(1)

We have equation of circle ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

We know that the perpendicular distance from centre to the tangent is equal to radius of the circle.

$L_{1} : x - y + 2 = 0;$ $L_{2} : 3 x - 4 y + 6 = 0;$ $L_{3} : 4 x - 3 y + 1 = 0$

So, $r = \frac{h - k + 2}{\sqrt{2}} = \frac{3 h - 4 k + 6}{\sqrt{9 + 16}} = \frac{4 h - 3 k + 1}{\sqrt{16 + 9}}$

Consider, $\frac{3 h - 4 k + 6}{5} = \frac{4 h - 3 k + 1}{5}$

$\Rightarrow 3 h - 4 k + 6 = 4 h - 3 k + 1 \Rightarrow h + k = 5$

Q 20 :

Let a circle $C_{1}$ be obtained on rolling the circle $x^{2} + y^{2} - 4 x - 6 y + 11 = 0$ upwards 4 units on the tangent T to it at the point (3, 2). Let $C_{2}$ be the image of $C_{1}$ in T. Let A and B be the centers of circles $C_{1}$ and $C_{2}$ respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x-axis. Then the area of the trapezium AMNB is: [2023]

$2 (2 + \sqrt{2})$
$2 (1 + \sqrt{2})$
$4 (1 + \sqrt{2})$
$3 + 2 \sqrt{2}$

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(3)

We have, $x^{2} + y^{2} - 4 x - 6 y + 11 = 0$ ...(i)

Centre = (2, 3); Radius $= \sqrt{2}$

Tangent at $(3, 2)$ is, $3 x + 2 y - 2 (x + 3) - 3 (y + 2) + 11 = 0$

$\Rightarrow x - y = 1$

On rolling the given circle (i) upwards 4 units on the tangent $x - y - 1 = 0$ , centre of the circle also moves upwards 4 units on $T$ . Let the centre of the new circle $C_{1}$ be $(h, k)$ .

Since, slope of tangent = Slope of line joining two centres of the circle.

Then the increment in both coordinates will be same.

From figure,

${(2 + a - 2)}^{2} + {(3 + a - 3)}^{2} = 16$

$a^{2} + a^{2} = 16 \Rightarrow a^{2} = 8 \Rightarrow a = 2 \sqrt{2}$

Hence, the centre of circle $C_{1}$ is $(2 + 2 \sqrt{2}, 3 + 2 \sqrt{2})$ and radius remains same $\sqrt{2}$ .

Equation of circle $C_{1}$ is ${(x - 2 - 2 \sqrt{2})}^{2} + {(y - 3 - 2 \sqrt{2})}^{2} = 2$

The centre of circle $C_{2}$ is the image of $C_{1}$ in $T$ , then

$\frac{x - 2 - \sqrt{2}}{1} = \frac{y - 3 - 2 \sqrt{2}}{- 1} = \frac{- 2 (2 + 2 \sqrt{2} - 3 - 2 \sqrt{2} - 1)}{1^{2} + {(- 1)}^{2}}$

$\Rightarrow x - 2 - 2 \sqrt{2} = 2 and y - 3 - 2 \sqrt{2} = - 2$

$\Rightarrow y = 1 + 2 \sqrt{2} \Rightarrow x = 4 + 2 \sqrt{2}$

The centre of circle $C_{2}$ is $B (4 + 2 \sqrt{2}, 1 + 2 \sqrt{2}) .$

Feet of perpendicular from $A$ and $B$ on the $x$ -axis are $M (2 + 2 \sqrt{2}, 0) and N (4 + 2 \sqrt{2}, 0)$ respectively.

Area of trapezium AMNB $= \frac{1}{2} (3 + 2 \sqrt{2} + 1 + 2 \sqrt{2}) (4 + 2 \sqrt{2} - 2 - 2 \sqrt{2})$

$= \frac{1}{2} \times (4 + 4 \sqrt{2}) \times 2 = 4 + 4 \sqrt{2} sq. units .$

Q 21 :

The set of all values of $a^{2}$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P (\frac{1 + a}{2}, \frac{1 - a}{2})$ on the circle $2 x^{2} + 2 y^{2} - (1 + a) x - (1 - a) y = 0,$ is equal to [2023]

$(8, \infty)$
$(0, 4]$
$(4, \infty)$
$(2, 12]$

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Q 22 :

Let the point $(p, p + 1)$ lie inside the region $E = {(x, y) : 3 - x \leq y \leq \sqrt{9 - x^{2}}, 0 \leq x \leq 3} .$ If the set of all values of $p$ is the interval $(a, b),$ then $b^{2} + b - a^{2}$ is equal to _____ . [2023]

(3)

$Given, 3 - x \leq y \leq \sqrt{9 - x^{2}}$

$3 - x \leq y \Rightarrow x + y - 3 \geq 0 \dots (i)$

$y \leq \sqrt{9 - x^{2}} \Rightarrow x^{2} + y^{2} \leq 9$

$Now, (p, p + 1) lies inside the region E .$ $∴$ $p + p + 1 - 3 \geq 0$

$\Rightarrow p \geq 1 and p^{2} + {(p + 1)}^{2} \leq 9$

$\Rightarrow 2 p^{2} + 2 p - 8 \leq 0 \Rightarrow p^{2} + p - 4 \leq 0$

$\Rightarrow p \in (- \frac{(1 + \sqrt{17})}{2}, \frac{\sqrt{17} - 1}{2})$

$∴$ $p \in (1, \frac{\sqrt{17} - 1}{2}) (as p \geq 1)$

$a = 1, b = \frac{\sqrt{17} - 1}{2} \Rightarrow b^{2} + b - a^{2} = 3$

Q 23 :

A circle passing through the point $P (α, β)$ in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of $α β$ is ________. [2023]

(121)

$Let the equation of the circle be$

${(x - a)}^{2} + {(y - a)}^{2} = a^{2},$

It passes through $P (α, β)$

$∴$ ${(α - a)}^{2} + {(β - a)}^{2} = a^{2}$

$\Rightarrow α^{2} + β^{2} - 2 a α - 2 a β + a^{2} = 0$ ...(i)

Here, the equation of line AB is $x + y = a$

Let $Q (α', β')$ be the foot of the perpendicular from $P$ on AB

$∴$ $\frac{α' - α}{1} = \frac{β' - β}{1} = \frac{- (α + β - a)}{2}$

${(P Q)}^{2} = {(α' - α)}^{2} + {(β' - β)}^{2} = \frac{1}{4} {(α + β - a)}^{2} + \frac{1}{4} {(α + β - a)}^{2}$

$\Rightarrow {(11)}^{2} = \frac{1}{2} {(α + β - a)}^{2}$

$\Rightarrow 242 = α^{2} + β^{2} + a^{2} + 2 α β - 2 a α - 2 a β$

$\Rightarrow 242 = 2 α β$ [From (i)]

$\Rightarrow α β = 121$

Q 24 :

Consider a circle $C_{1} : x^{2} + y^{2} - 4 x - 2 y = α - 5$ . Let its mirror image in the line $y = 2 x + 1$ be another circle $C_{2} : 5 x^{2} + 5 y^{2} - 10 f x - 10 g y + 36 = 0$ . Let $r$ be the radius of $C_{2}$ . Then $α + r$ is equal to _______ . [2023]

(2)

We have,

$C_{1} : x^{2} + y^{2} - 4 x - 2 y - (α - 5) = 0$

$∴$ $Centre is (2, 1), r = \sqrt{4 + 1 + α - 5} = \sqrt{α}$

$C_{2} : 5 x^{2} + 5 y^{2} - 10 f x - 10 g y + 36 = 0$

$i.e., x^{2} + y^{2} - 2 f x - 2 g y + \frac{36}{5} = 0$

$Centre is (f, g), r = \sqrt{f^{2} + g^{2} - \frac{36}{5}}$

$The image of (2, 1) with respect to 2 x - y + 1 = 0 is (f, g),$

$then \frac{f - 2}{2} = \frac{g - 1}{- 1} = \frac{- 2 (4 - 1 + 1)}{5}$

$\Rightarrow \frac{f - 2}{2} = \frac{- 8}{5} and g - 1 = \frac{8}{5}$

$\Rightarrow f - 2 = - \frac{16}{5} and g - 1 = \frac{8}{5}$

$\Rightarrow f = 2 - \frac{16}{5} = \frac{- 6}{5}$ and $g = \frac{13}{5}$

$So, (f, g) = (- \frac{6}{5}, \frac{13}{5})$

$and r = \sqrt{\frac{36}{25} + \frac{169}{25} - \frac{36}{5}} \Rightarrow r = 1$

$\Rightarrow r = 1$

$∴$ $α + r = 1 + 1 = 2$

Q 25 :

Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$ . Then $r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}$ is equal to _______ . [2023]

(7)

Q 26 :

Points P(-3, 2), Q(9, 10) and R( $α$ , 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $2 x - k y = 1$ , then $k$ is equal to ______ . [2023]

(3)

$Since P Q ⊥ Q R, so m_{P Q} \cdot m_{Q R} = - 1$

$\Rightarrow \frac{10 - 2}{9 + 3} \times \frac{10 - 4}{9 - α} = - 1 \Rightarrow α = 13$

$Since Q O ⊥ Q S, so m_{O Q} \cdot m_{Q S} = - 1 \Rightarrow m_{Q S} = - \frac{4}{7}$

$Equation of Q S : y - 10 = - \frac{4}{7} (x - 9) \Rightarrow 4 x + 7 y = 106 (i)$

$Since O R ⊥ S R$

$m_{O R} \cdot m_{R S} = - 1$

$\Rightarrow m_{R S} = - 8$

$Equation of R S : y - 4 = - 8 (x - 13)$

$\Rightarrow 8 x + y = 108 (ii)$

$Solving (i) and (ii)$

$x = \frac{25}{2}, y = 8$

$Since S (x, y) lies on the line 2 x - k y = 1$

$25 - 8 k = 1 \Rightarrow 8 k = 24 \Rightarrow k = 3$

Q 27 :

A circle with centre (2, 3) and radius 4 intersects the line $x + y = 3$ at the points P and Q. If the tangents at P and Q intersect at the point $S (α, β)$ , then $4 α - 7 β$ is equal to ______ . [2023]

(11)

$The equation of circle can be written as,$

${(x - 2)}^{2} + {(y - 3)}^{2} = 4^{2} \Rightarrow x^{2} + y^{2} - 4 x - 6 y - 3 = 0$

$Chord of contact at (α, β) is$

$\Rightarrow α x + β y - 2 (x + α) - 3 (y + β) - 3 = 0$

$\Rightarrow (α - 2) x + (β - 3) y - (2 α + 3 β + 3) = 0 ...(i)$

$∵$ $But the equation of chord of contact is given:$

$x + y - 3 = 0 ...(ii)$

$Comparing equations (i) and (ii):$

$\frac{α - 2}{1} = \frac{β - 3}{1} = - (\frac{2 α + 3 β + 3}{- 3})$

$On solving, α = - 6, β = - 5$

$So, 4 α - 7 β = 4 (- 6) - 7 (- 5) = 11$

Q 28 :

Let $P (a_{1}, b_{1})$ and $Q (a_{2}, b_{2})$ be two distinct points on a circle with center $C (\sqrt{2}, \sqrt{3})$ . Let O be the origin and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is $\frac{\sqrt{35}}{2}$ , then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to ________ . [2023]

(24)

$Area of ∆ O C P = \frac{\sqrt{35}}{2}$

$\frac{1}{2} \times P C \times O C = \frac{\sqrt{35}}{2}$

$\frac{1}{2} \times P C \times \sqrt{5} = \frac{\sqrt{35}}{2}$

$P C = \sqrt{7}$

$Now, O Q^{2} = 5 + 7 = 12$

$O P^{2} = 5 + 7 = 12$

$a_{1}^{2} + b_{1}^{2} = O P^{2} and a_{2}^{2} + b_{2}^{2} = O Q^{2}$

$∴$ $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2} = O P^{2} + O Q^{2} = 12 + 12 = 24$

Q 29 :

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and angle $∠ A O B = \frac{π}{3}$ Then the locus of the point of intersection of the lines PQ and MN is: [2026]

$3 (x^{2} + y^{2}) - 12 x - 18 y - 25 = 0$
$3 (x^{2} + y^{2}) - 18 x - 12 y + 25 = 0$
$x^{2} + y^{2} - 12 x - 18 y - 25 = 0$
$x^{2} + y^{2} - 18 x - 12 y - 25 = 0$

1

2

3

4

(1)

Given Circle

$x^{2} + y^{2} - 4 x - 6 y - 3 = 0$

$C (2, 3) and r = 4$

$\cos 30 ° = \frac{r}{O R} = \frac{4}{O R}$

$\Rightarrow O R = \frac{8}{\sqrt{3}}$

Now

$O R^{2} = {(h - 2)}^{2} + {(k - 3)}^{2}$

$\Rightarrow 3 (x^{2} + y^{2}) - 12 x - 18 y - 25 = 0$

Q 30 :

Let a circle of radius 4 pass through the origin O, the points $A (- \sqrt{3} a, 0)$ and $B (0, - \sqrt{2} b),$ where $a$ and $b$ are real parameters and $a b \neq 0$ . Then the locus of the centroid of $∆ O A B$ is a circle of radius [2026]

$\frac{7}{3}$
$\frac{8}{3}$
$\frac{5}{3}$
$\frac{11}{3}$

1

2

3

4

(2)

$A B = 8$

$3 a^{2} + 2 b^{2} = 64$

$Centroid G (h, k)$

$h = - \frac{\sqrt{3 a}}{3}, k = - \frac{\sqrt{2 b}}{3}$

$a = - \sqrt{3} h, b = - \frac{3}{\sqrt{2}} k$

$9 h^{2} + 9 k^{2} = 64$

$x^{2} + y^{2} = \frac{64}{9}$

$r = \frac{8}{3}$

Q 31 :

Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

24
21
20
25

1

2

3

4

(4)

${(x - 2)}^{2} + {(y - 1)}^{2} = 3^{2} & {(x + 1)}^{2} + {(y + 4)}^{2} = r^{2}$

$| r_{1} - r_{2} |$ $< c_{1} c_{2} < r_{1} + r_{2}$

$| r - 3 |$ $< \sqrt{{(2 + 1)}^{2} + {(1 + 4)}^{2}} < r + 3$

$| r - 3 |$ $< \sqrt{34} & r + 3 > \sqrt{34}$

$- \sqrt{34} < r - 3 < \sqrt{34} & r > \sqrt{34} - 3$

$i.e. r = (3 - \sqrt{34}, 3 + \sqrt{34}) \cap (\sqrt{34} - 3, \infty)$

$i.e. r \in (\sqrt{34} - 3, \sqrt{34} + 3)$

$∴$ $α β = (\sqrt{34} - 3) (\sqrt{34} + 3)$

$= 34 - 9$

$= 25$

Q 32 :

Let the circle $x^{2} + y^{2} = 4$ intersect x-axis at the points $A (a, 0), a > 0 and B (b, 0) .$ Let $P (2 \cos α, 2 \sin α) 0 < α < \frac{π}{2} and Q (2 \cos β, 2 \sin β)$ be two points such that $(α - β) = \frac{π}{2}$ . Then the point of intersection of AQ and BP lies on : [2026]

$x^{2} + y^{2} - 4 x - 4 = 0$
$x^{2} + y^{2} - 4 y - 4 = 0$
$x^{2} + y^{2} - 4 x - 4 y - 4 = 0$
$x^{2} + y^{2} - 4 x - 4 y = 0$

1

2

3

4

(2)

$Let point of intersection R (h, k)$

$m_{B R} = m_{B P} \Rightarrow \frac{k}{h + 2} = \frac{2 \sin α}{2 \cos α + 2} \Rightarrow \frac{k}{h + 2} = \tan \frac{α}{2}$

$m_{A R} = m_{A Q} \Rightarrow \frac{k}{h - 2} = \frac{2 \sin β}{2 \cos β - 2} = \frac{\sin β}{\cos β - 1} = - \cot \frac{β}{2}$

$\frac{α}{2} - \frac{β}{2} = \frac{π}{4}$

$\tan (\frac{α}{2} - \frac{β}{2}) = \tan \frac{π}{4} = 1$

$\frac{\tan \frac{α}{2} - \tan \frac{β}{2}}{1 + \tan \frac{α}{2} \tan \frac{β}{2}} = 1$

$\frac{\frac{k}{h + 2} + \frac{h - 2}{k}}{1 + (\frac{k}{h + 2}) (\frac{2 - h}{k})} = 1$

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

$\Rightarrow \frac{h^{2} + k^{2} - 4}{4 k} = 1$

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

Q 33 :

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

(9)

$Let P \equiv (2 \cos θ, 2 \sin θ)$

$∴$ $coordinates of Q = (4 \cos θ + 1, 6 \sin θ + 3)$

$∴$ $locus of Q is {(\frac{x - 1}{4})}^{2} + {(\frac{y - 3}{6})}^{2} = 1$

$∴$ $e^{2} = 1 - \frac{16}{36} = \frac{5}{9}$

$∴$ $\frac{5}{e^{2}} = 9$

Q 34 :

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

(2)

Mid point of $P Q$ lies on $x - y + 1 = 0$

$\frac{2 \cos θ + α}{2} - \frac{2 \sin θ - 5 α - 2}{2} + 1 = 0$

$2 \cos θ + α - 2 \sin θ + 5 α + 2 + 2 = 0$

$\cos θ - \sin θ + 3 α + 2 = 0 . . . (1)$

$∵$ $Slope of P Q is - 1$

$\frac{2 \sin θ + 5 α + 2}{2 \cos θ - α} = - 1$

$2 \sin θ + 5 α + 2 = - 2 \cos θ + α$

$\sin θ + \cos θ + 2 α + 1 = 0 . . . (2)$

Eliminate $α$ from (1) and (2)

$\Rightarrow \cos θ + 5 \sin θ = 1, θ \in [0, 2 π]$

$\Rightarrow 5 \times 2 \sin \frac{θ}{2} \cos \frac{θ}{2} = 2 \sin^{2} \frac{θ}{2}$

$∴$ $\sin \frac{θ}{2} = 0 \Rightarrow \cos θ = 1$

or

$\sin \frac{θ}{2} = 5 \Rightarrow \cos θ = - \frac{12}{13}$

Sum of all possible values of abscissa of point $P$ is

$= 2 \times 1 + 2 (\frac{- 12}{13}) = \frac{2}{13}$

$∴$ $13 times sum of all possible values of abscissa of point P is = 2$

Q 35 :

Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$ , which passes through the point (2,3) and is farthest from the center of $C_{2}$ , is $x + a y + b = 0$ , then a−b is equal to [2026]

10
-2
6
-6

1

2

3

4

(2)

$Equation of circle C_{2} is$

$x^{2} + y^{2} - 5 x - 5 y = 0$

$Its centre is (\frac{5}{2}, \frac{5}{2})$

$m_{A B} = - 1$

$∴$ $Slope of required chord = 1$

$∴$ $Equation of required chord is x - y + 1 = 0$

$∴$ $a = - 1, b = 2$

$∴$ $a - b = - 2$

Topic Question Set

Q 11 :

A line segment AB of length $λ$ moves such that the points A and B remain on the periphery of a circle of radius $λ$ . Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

Q 12 :

Let A be the point (1, 2) and B be any point on the curve $x^{2} + y^{2} = 16$ . If the centre of the locus of the point P, which divides the line segment AB in the ratio 3 : 2, is the point $C (α, β)$ , then the length of the line segment AC is [2023]

Q 13 :

Let the centre of a circle C be $(α, β)$ and its radius $r < 8$ . Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to C. Then $(α - β + r)$ is equal to [2023]

Q 14 :

The number of common tangents to the circles $x^{2} + y^{2} - 18 x - 15 y + 131 = 0$ and $x^{2} + y^{2} - 6 x - 6 y - 7 = 0$ , is [2023]

Q 17 :

The points of intersection of the line $a x + b y = 0 (a \neq b)$ and the circle $x^{2} + y^{2} - 2 x = 0$ are $A (α, 0)$ and $B (1, β)$ . The image of the circle with $A B$ as a diameter in the line $x + y + 2 = 0$ is [2023]

Q 18 :

Let the tangents at the points $A (4, - 11)$ and $B (8, - 5)$ on the circle $x^{2} + y^{2} - 3 x + 10 y - 15 = 0,$ intersect at the point $C$ . Then the radius of the circle whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to [2023]

Q 19 :

Let $y = x + 2, 4 y = 3 x + 6$ and $3 y = 4 x + 1$ be three tangent lines to the circle ${(x - h)}^{2} + {(y - k)}^{2} = r^{2} .$ Then $h + k$ is equal to [2023]

Q 21 :

The set of all values of $a^{2}$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P (\frac{1 + a}{2}, \frac{1 - a}{2})$ on the circle $2 x^{2} + 2 y^{2} - (1 + a) x - (1 - a) y = 0,$ is equal to [2023]

(1)

Q 22 :

Let the point $(p, p + 1)$ lie inside the region $E = {(x, y) : 3 - x \leq y \leq \sqrt{9 - x^{2}}, 0 \leq x \leq 3} .$ If the set of all values of $p$ is the interval $(a, b),$ then $b^{2} + b - a^{2}$ is equal to _____ . [2023]

Q 24 :

Consider a circle $C_{1} : x^{2} + y^{2} - 4 x - 2 y = α - 5$ . Let its mirror image in the line $y = 2 x + 1$ be another circle $C_{2} : 5 x^{2} + 5 y^{2} - 10 f x - 10 g y + 36 = 0$ . Let $r$ be the radius of $C_{2}$ . Then $α + r$ is equal to _______ . [2023]

Q 25 :

Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$ . Then $r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}$ is equal to _______ . [2023]

(7)

Q 26 :

Points P(-3, 2), Q(9, 10) and R( $α$ , 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $2 x - k y = 1$ , then $k$ is equal to ______ . [2023]

Q 27 :

A circle with centre (2, 3) and radius 4 intersects the line $x + y = 3$ at the points P and Q. If the tangents at P and Q intersect at the point $S (α, β)$ , then $4 α - 7 β$ is equal to ______ . [2023]

Q 29 :

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and angle $∠ A O B = \frac{π}{3}$ Then the locus of the point of intersection of the lines PQ and MN is: [2026]

Q 30 :

Let a circle of radius 4 pass through the origin O, the points $A (- \sqrt{3} a, 0)$ and $B (0, - \sqrt{2} b),$ where $a$ and $b$ are real parameters and $a b \neq 0$ . Then the locus of the centroid of $∆ O A B$ is a circle of radius [2026]

Q 31 :

Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

Q 33 :

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

Q 34 :

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

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Topic Question Set

Q 11 : A line segment AB of length λ moves such that the points A and B remain on the periphery of a circle of radius λ. Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

Q 12 : Let A be the point (1, 2) and B be any point on the curve x2+y2=16. If the centre of the locus of the point P, which divides the line segment AB in the ratio 3 : 2, is the point C(α,β), then the length of the line segment AC is [2023]

Q 13 : Let the centre of a circle C be (α,β) and its radius r<8. Let 3x+4y=24 and 3x-4y=32 be two tangents and 4x+3y=1 be a normal to C. Then (α-β+r) is equal to [2023]

Q 14 : The number of common tangents to the circles x2+y2-18x-15y+131=0 and x2+y2-6x-6y-7=0, is [2023]

Q 16 : The locus of the mid points of the chords of the circle C1:(x-4)2+(y-5)2=4 which subtend an angle θi at the centre of the circle C1, is a circle of radius ri. If θ1=π3,θ3=2π3 and r12=r22+r32, then θ2 is equal to [2023]

Q 17 : The points of intersection of the line ax+by=0(a≠b) and the circle x2+y2-2x=0 are A(α,0) and B(1,β). The image of the circle with AB as a diameter in the line x+y+2=0 is [2023]

Q 18 : Let the tangents at the points A(4,-11) and B(8,-5) on the circle x2+y2-3x+10y-15=0, intersect at the point C. Then the radius of the circle whose centre is C and the line joining A and B is its tangent, is equal to [2023]

Q 19 : Let y=x+2, 4y=3x+6 and 3y=4x+1 be three tangent lines to the circle (x-h)2+(y-k)2=r2. Then h+k is equal to [2023]

Q 21 : The set of all values of a2 for which the line x+y=0 bisects two distinct chords drawn from a point P(1+a2,1-a2) on the circle 2x2+2y2-(1+a)x-(1-a)y=0,is equal to [2023]

(1)

Q 22 : Let the point (p, p+1) lie inside the region E={(x,y):3-x≤y≤9-x2, 0≤x≤3}. If the set of all values of p is the interval (a,b), then b2+b-a2 is equal to _____ . [2023]

Q 24 : Consider a circle C1:x2+y2-4x-2y=α-5. Let its mirror image in the line y=2x+1 be another circle C2:5x2+5y2-10fx-10gy+36=0. Let r be the radius of C2. Then α+r is equal to _______ . [2023]

Q 25 : Two circles in the first quadrant of radii r1 and r2 touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x+y=2. Then r12+r22-r1r2 is equal to _______ . [2023]

(7)

Q 26 : Points P(-3, 2), Q(9, 10) and R(α, 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2x-ky=1, then k is equal to ______ . [2023]

Q 27 : A circle with centre (2, 3) and radius 4 intersects the line x+y=3 at the points P and Q. If the tangents at P and Q intersect at the point S(α,β), then 4α-7β is equal to ______ . [2023]

Q 28 : Let P(a1,b1) and Q(a2,b2) be two distinct points on a circle with center C(2,3). Let O be the origin and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is 352, then a12+a22+b12+b22 is equal to ________ . [2023]

Q 29 : Let PQ and MN be two straight lines touching the circle x2+y2-4x-6y-3=0 at the points A and B respectively. Let O be the centre of the circle and angle ∠AOB=π3 Then the locus of the point of intersection of the lines PQ and MN is: [2026]

Q 30 : Let a circle of radius 4 pass through the origin O, the points A(-3a,0) and B(0,-2b), where a and b are real parameters and ab≠0. Then the locus of the centroid of ∆OAB is a circle of radius [2026]

Q 31 : Let the set of all values of r, for which the circles (x+1)2+(y+4)2=r2 and x2+y2-4x-2y-4=0 intersect at two distinct points be the interval (α,β). Then αβ is equal to [2026]

Q 32 : Let the circle x2+y2=4 intersect x-axis at the points A(a,0), a>0 and B(b,0). Let P(2cosα, 2sinα) 0<α<π2 and Q(2cosβ, 2sinβ) be two points such that (α-β)=π2. Then the point of intersection of AQ and BP lies on : [2026]

Q 33 : Let (h,k) lie on the circle C:x2+y2=4 and the point (2h+1, 3k+2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to _________ [2026]

Q 34 : If P is a point on the circle x2+y2=4,, Q is a point on the straight line 5x+y+2=0 and x−y+1=0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

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Q 11 :

A line segment AB of length $λ$ moves such that the points A and B remain on the periphery of a circle of radius $λ$ . Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

Q 12 :

Let A be the point (1, 2) and B be any point on the curve $x^{2} + y^{2} = 16$ . If the centre of the locus of the point P, which divides the line segment AB in the ratio 3 : 2, is the point $C (α, β)$ , then the length of the line segment AC is [2023]

Q 13 :

Let the centre of a circle C be $(α, β)$ and its radius $r < 8$ . Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to C. Then $(α - β + r)$ is equal to [2023]

Q 14 :

The number of common tangents to the circles $x^{2} + y^{2} - 18 x - 15 y + 131 = 0$ and $x^{2} + y^{2} - 6 x - 6 y - 7 = 0$ , is [2023]

Q 17 :

The points of intersection of the line $a x + b y = 0 (a \neq b)$ and the circle $x^{2} + y^{2} - 2 x = 0$ are $A (α, 0)$ and $B (1, β)$ . The image of the circle with $A B$ as a diameter in the line $x + y + 2 = 0$ is [2023]

Q 18 :

Let the tangents at the points $A (4, - 11)$ and $B (8, - 5)$ on the circle $x^{2} + y^{2} - 3 x + 10 y - 15 = 0,$ intersect at the point $C$ . Then the radius of the circle whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to [2023]

Q 19 :

Let $y = x + 2, 4 y = 3 x + 6$ and $3 y = 4 x + 1$ be three tangent lines to the circle ${(x - h)}^{2} + {(y - k)}^{2} = r^{2} .$ Then $h + k$ is equal to [2023]

Q 21 :

The set of all values of $a^{2}$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P (\frac{1 + a}{2}, \frac{1 - a}{2})$ on the circle $2 x^{2} + 2 y^{2} - (1 + a) x - (1 - a) y = 0,$ is equal to [2023]

Q 22 :

Let the point $(p, p + 1)$ lie inside the region $E = {(x, y) : 3 - x \leq y \leq \sqrt{9 - x^{2}}, 0 \leq x \leq 3} .$ If the set of all values of $p$ is the interval $(a, b),$ then $b^{2} + b - a^{2}$ is equal to _____ . [2023]

Q 24 :

Consider a circle $C_{1} : x^{2} + y^{2} - 4 x - 2 y = α - 5$ . Let its mirror image in the line $y = 2 x + 1$ be another circle $C_{2} : 5 x^{2} + 5 y^{2} - 10 f x - 10 g y + 36 = 0$ . Let $r$ be the radius of $C_{2}$ . Then $α + r$ is equal to _______ . [2023]

Q 25 :

Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$ . Then $r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}$ is equal to _______ . [2023]

Q 26 :

Points P(-3, 2), Q(9, 10) and R( $α$ , 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $2 x - k y = 1$ , then $k$ is equal to ______ . [2023]

Q 27 :

A circle with centre (2, 3) and radius 4 intersects the line $x + y = 3$ at the points P and Q. If the tangents at P and Q intersect at the point $S (α, β)$ , then $4 α - 7 β$ is equal to ______ . [2023]

Q 29 :

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and angle $∠ A O B = \frac{π}{3}$ Then the locus of the point of intersection of the lines PQ and MN is: [2026]

Q 30 :

Let a circle of radius 4 pass through the origin O, the points $A (- \sqrt{3} a, 0)$ and $B (0, - \sqrt{2} b),$ where $a$ and $b$ are real parameters and $a b \neq 0$ . Then the locus of the centroid of $∆ O A B$ is a circle of radius [2026]

Q 31 :

Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

Q 33 :

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

Q 34 :

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]