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

Q 11 :

Let a tangent to the curve $y^{2} = 24 x$ meet the curve $x y = 2$ at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

length of latus rectum 3/2
length of latus rectum 2
directrix $4 x$ = 3
directrix $4 x$ = - 3

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(3)

$Let the equation of tangent to y^{2} = 24 x$ $is t y = x + 6 t^{2}$

$Now, t y = x + 6 t^{2} meets the curve x y = 2 at points A and B .$

$Let mid-point of A B be (h, k)$

$t y = \frac{2}{y} + 6 t^{2}$ $[∵ x = \frac{2}{y}]$

$t y^{2} - 6 t^{2} y - 2 = 0$

$y_{1} + y_{2} = 6 t$

$t \cdot \frac{2}{x} = x + 6 t^{2}$ $[∵ y = \frac{2}{x}]$

$x^{2} + 6 t^{2} x - 2 t = 0$

$x_{1} + x_{2} = - 6 t^{2}$

$Midpoint P is (- 3 t^{2}, 3 t)$

$\Rightarrow h = - 3 t^{2}, k = 3 t$ $\Rightarrow (\frac{h}{- 3}) = {(\frac{k}{3})}^{2} \Rightarrow y^{2} = - 3 x$

Q 12 :

The equations of the sides AB and AC of a triangle ABC are $(λ + 1) x + λ y = 4 and λ x + (1 - λ) y + λ = 0$ respectively. Its vertex A is on the $y$ -axis and its orthocentre is (1, 2). The length of the tangent from the point C to the be part of the parabola $y^{2} = 6 x$ in the first quadrant is [2023]

$2 \sqrt{2}$
$2$
$\sqrt{6}$
$4$

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(1)

$A B : (λ + 1) x + λ y = 4$

$A C : λ x + (1 - λ) y + λ = 0$

Since vertex $A$ is on $y$ -axis $\Rightarrow x = 0$

$So, y = \frac{4}{λ} and y = \frac{λ}{λ - 1}$
$\Rightarrow \frac{4}{λ} = \frac{λ}{λ - 1} \Rightarrow λ = 2$

$A B : 3 x + 2 y = 4$

$A C : 2 x - y + 2 = 0 \Rightarrow A (0, 2)$

Let coordinates of $C$ be $(α, 2 α + 2)$ .

$Now, [slope of altitude through C] \times (\frac{- 3}{2}) = - 1$

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

So, coordinates of $C$ are $(\frac{- 1}{2}, 1) .$

Equation of tangent be $y = m x + \frac{3}{2 m}$

$∴$ $m = 1, - 3$

So, tangent which touches in first quadrant at $T$ is

$T \equiv (\frac{a}{m^{2}}, \frac{2 a}{m}) \equiv (\frac{3}{2}, 3)$

$∴$ $C T = \sqrt{{(\frac{3}{2} + \frac{1}{2})}^{2} + {(3 - 1)}^{2}} = \sqrt{4 + 4} = 2 \sqrt{2}$

Q 13 :

The equations of two sides of a variable triangle are $x = 0$ and $y = 3,$ and its third side is a tangent to the parabola $y^{2} = 6 x$ . The locus of its circumcentre is [2023]

$4 y^{2} - 18 y + 3 x + 18 = 0$
$4 y^{2} - 18 y - 3 x + 18 = 0$
$4 y^{2} + 18 y + 3 x + 18 = 0$
$4 y^{2} - 18 y - 3 x - 18 = 0$

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(1)

$Equation of the parabola is y^{2} = 6 x .$

Here $4 a = 6$

$a = \frac{3}{2}$

Equation of tangent is $y = m x + \frac{a}{m}$

$\Rightarrow y = m x + \frac{3}{2 m}$ ...(i)

Putting $x = 0$ in (i), we get $A \equiv (0, \frac{3}{2 m})$

Putting $y = 3$ in (i), we get $B \equiv (\frac{6 m - 3}{2 m^{2}}, 3)$

The centre of the circle will lie on the line $A B$ as midpoint.

$∴$ $h = \frac{6 m - 3}{4 m^{2}}, k = \frac{3 + 6 m}{4 m}$ $\Rightarrow m = \frac{3}{4 k - 6}$

On substituting $h = \frac{6 m - 3}{4 m^{2}}$ to eliminate $m$ , we get $3 h = 2 (- 2 k^{2} + 9 k - 9)$

$4 k^{2} - 18 k + 3 h + 18 = 0$

So, locus is $4 y^{2} - 18 y + 3 x + 18 = 0$

Q 14 :

If the tangent at a point P on the parabola $y^{2} = 3 x$ is parallel to the line $x + 2 y = 1$ and the tangents at the points Q and R on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$ are perpendicular to the line $x - y = 2$ , then the area of the triangle PQR is [2023]

$3 \sqrt{5}$
$\frac{9}{\sqrt{5}}$
$\frac{3}{2} \sqrt{5}$
$5 \sqrt{3}$

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(1)

If tangent at a point $P$ on $y^{2} = 3 x$ is parallel to the line $x + 2 y = 1$ and tangent at point $Q$ and $R$ on ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$ are perpendicular to the line $x - y = 2$ .

Firstly we have equation of parabola $y^{2} = 3 x \dots (i)$

Tangent at $P (x_{1}, y_{1})$ is parallel to $x + 2 y = 1$

$2 y = 1 - x$

$y = - \frac{x}{2} + \frac{1}{2}$

{On comparing it with $y = m x + C$ }

Then slope, $m$ at $P = - \frac{1}{2} \dots (i i)$

On differentiating equation (i) with respect to $' x'$

$2 y \cdot \frac{d y}{d x} = 3 \Rightarrow \frac{d y}{d x} = \frac{3}{2 y} = - \frac{1}{2} (from (ii))$

$\Rightarrow y_{1} = - 3$

Co-ordinates of $P$ are $(3, - 3)$ .

Similarly, $Q (\frac{4}{\sqrt{5}}, \frac{1}{\sqrt{5}}), R (\frac{- 4}{\sqrt{5}}, \frac{- 1}{\sqrt{5}})$

So, we have three points P, Q and R by which area of $∆ P Q R$

$= \frac{1}{2}$ $| \begin{matrix} 3 & - 3 & 1 \\ \frac{4}{\sqrt{5}} & \frac{1}{\sqrt{5}} & 1 \\ \frac{- 4}{\sqrt{5}} & \frac{- 1}{\sqrt{5}} & 1 \end{matrix} |$ ${∴ Area of ∆ is given as = \frac{1}{2} | \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} |}$

$= \frac{30}{2 \sqrt{5}} = 3 \sqrt{5}$

Hence, the area of $∆ P Q R = 3 \sqrt{5}$ .

Q 15 :

If $P (h, k)$ be a point on the parabola $x = 4 y^{2}$ , which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola $y^{2} = 4 (x + y)$ is equal to [2023]

4
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8
6

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(4)

We have $x = 4 y^{2}$

Equation of normal at $P (h, k)$

$y - k = - \frac{k}{(\frac{1}{8})} (x - h)$

$y - k = - 8 k (x - h)$

This line passes through the point $(0, 33)$ .

$33 - k = 8 k h$ ...(i)

$P (h, k)$ will satisfy $x = 4 y^{2}$ , so $h = 4 k^{2}$ . ...(ii)

Solving equations (i) and (ii), we get

$k = 1 and h = 4$

$∴$ $(h, k) \equiv (4, 1)$

Now we have equation of parabola, $y^{2} = 4 (x + y) \Rightarrow {(y - 2)}^{2} = 4 (x + 1)$

The directrix of this parabola is $x = - 2$

This is the line parallel to the y-axis.

So, distance of $P (4, 1)$ from the line $x = - 2$ is $6$ .

Q 16 :

The parabolas : $a x^{2} + 2 b x + c y = 0$ and $d x^{2} + 2 e x + f y = 0$ intersect on the line $y = 1$ . If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then [2023]

$\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in G.P.$
$d, e, f are in A.P.$
$d, e, f are in G.P.$
$\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$

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(4)

$a x^{2} + 2 b x + c y = 0$ ...(i)

$and d x^{2} + 2 e x + f y = 0$ ...(ii)

Equation (i) and (ii) intersect at $(α, 1)$ .

$∴$ $a α^{2} + 2 b α + c = 0$ ...(iii)

$d α^{2} + 2 e α + f = 0$ ...(iv)

Roots of equation (iii):

$α = \frac{- 2 b \pm \sqrt{4 b^{2} - 4 a c}}{2 a} = - \frac{b}{a} [∵ a, b, c are in G.P. \Rightarrow b^{2} = a c]$

$α is also a root of equation (iv)$

$∴$ $d {(\frac{- b}{a})}^{2} + 2 e (\frac{- b}{a}) + f = 0$ $\Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{2 e}{b}$

$∴$ $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c} are in A.P.$

Q 17 :

Let A be a point on the x-axis. Common tangents are drawn from A to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at Q and R, then ${(Q R)}^{2}$ is equal to [2023]

76
81
72
64

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(3)

Given curves are $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$

Equation of tangent to the parabola in slope form is:

$y = m x + \frac{a}{x} \Rightarrow y = m x + \frac{4}{m}$

Length of perpendicular from (0, 0) to the point of tangency is equal to the length of radius of circle.

$\Rightarrow$ $| \frac{\frac{4}{m}}{\sqrt{1 + m^{2}}} |$ $= \sqrt{8} \Rightarrow$ $| \frac{1}{m \sqrt{1 + m^{2}}} |$ $= \frac{1}{\sqrt{2}}$

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

$(m^{2} + 2) (m^{2} - 1) = 0$ $∴$ $m = \pm 1$

Point of contact on parabola = $(\frac{a}{m^{2}}, \frac{2 a}{m}) = (4, \pm 8)$

Point of contact on circle is $(- 2, 2) or (2, - 2)$

Distance between $Q (- 2, 2)$ and $R (4, 8)$ is $Q R \Rightarrow Q R^{2} = 72$

Q 18 :

Let $y = f (x)$ represent a parabola with focus $(- \frac{1}{2}, 0)$ and directrix $y = - \frac{1}{2}$ . Then $S = {x \in ℝ : \tan^{- 1} (\sqrt{f (x)}) + \sin^{- 1} (\sqrt{f (x) + 1}) = \frac{π}{2}} :$ [2023]

is an empty set
contains exactly one element
contains exactly two elements
is an infinite set

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(3)

Equations of parabola, $\sqrt{{(x + \frac{1}{2})}^{2} + {(y - 0)}^{2}} =$ $| y + \frac{1}{2} |$ $\Rightarrow x^{2} + \frac{1}{4} + x + y^{2} = y^{2} + \frac{1}{4} + y$

$\Rightarrow x^{2} + x = y = f (x) (let)$

Now, $\tan^{- 1} \sqrt{f (x)} + \sin^{- 1} \sqrt{f (x) + 1} = \frac{π}{2}$

$\cos^{- 1} \frac{1}{\sqrt{1 + f (x)}} + \sin^{- 1} \sqrt{f (x) + 1} = \frac{π}{2}$

We know, $\sin^{- 1} x + \cos^{- 1} x = \frac{π}{2},$

$\Rightarrow \frac{1}{\sqrt{1 + f (x)}} = \sqrt{f (x) + 1} \Rightarrow f (x) + 1 = 1 \Rightarrow f (x) = 0$

$x^{2} + x = 0 \Rightarrow x (x + 1) = 0 \Rightarrow x = 0, - 1$ $\Rightarrow S = {- 1, 0}$

Q 19 :

Let the tangent to the curve $x^{2} + 2 x - 4 y + 9 = 0$ at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line $x - 3 y = 6$ meet the parabola $y^{2} = 4 x$ at B. If B lies on the line $2 x - 3 y = 8,$ then ${(A B)}^{2}$ is equal to ________ . [2023]

(292)

Equation of tangent at P(1, 3) to the curve $x^{2} + 2 x - 4 y + 9 = 0$ is

$x \cdot 1 + 2 (\frac{1 + x}{2}) - 4 (\frac{3 + y}{2}) + 9 = 0$

$\Rightarrow x + 1 + x - 6 - 2 y + 9 = 0$

$\Rightarrow 2 x - 2 y + 4 = 0 \Rightarrow y - x = 2$

So, the point $A$ is $(0, 2)$ .

Equation of line passing through P and parallel to $x - 3 y = 6$ is $x - 3 y = d$ .

Point $(1, 3)$ lies on this line.

$∴$ $1 - 9 = d \Rightarrow d = - 8$

So, equation of line is $x - 3 y = - 8$ .

Now, $x - 3 y = - 8$ meets the parabola $y^{2} = 4 x$ at $B$ .

So, $\frac{x + 8}{3} = y \Rightarrow {(\frac{x + 8}{3})}^{2} = 4 x$ $\Rightarrow x^{2} + 16 x + 64 = 36 x$

$\Rightarrow x^{2} - 16 x - 4 x + 64 = 0 \Rightarrow x = 4, 16$ $∴$ $y = 4, 8$

So, the possible coordinates of $B$ are $(4, 4)$ or $(16, 8)$ .

But $(4, 4)$ doesn't lie on the line $2 x - 3 y = 8$ .

Thus, point $B$ is $(16, 8)$ .

$∴$ $A B^{2} = {(16 - 0)}^{2} + {(8 - 2)}^{2} = 256 + 36 = 292$

Q 20 :

The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix $x = - 3$ are in the ratio 3 : 1. If $R (α, β)$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{β^{2}}{α}$ is equal to _______. [2023]

(16)

Parabola is $y^{2} = 12 x$

$Let Q (3 t^{2}, 6 t)$

$So, P (27 t^{2}, 18 t)$

$Now, R (α, β) = (a t_{1} t_{2}, a (t_{1} + t_{2}))$

$= (3 t \cdot 3 t, 3 (t + 3 t)) = (9 t^{2}, 12 t)$

$R (α, β) = (9 t^{2}, 12 t) \Rightarrow \frac{β^{2}}{α} = \frac{{(12 t)}^{2}}{9 t^{2}} = \frac{144}{9} = 16$

Q 21 :

Let a common tangent to the curves $y^{2} = 4 x$ and ${(x - 4)}^{2} + y^{2} = 16$ touch the curves at the points P and Q. Then ${(P Q)}^{2}$ is equal to _________ . [2023]

(32)

Equation of tangent to parabola $y^{2} = 4 x$ is given by

$y = m x + \frac{1}{m}$ ...(i)

and equation of tangent to the circle ${(x - 4)}^{2} + y^{2} = 16$ is given by

$y = m (x - 4) \pm 4 \sqrt{1 + m^{2}}$ ...(ii)

From (i) and (ii), we get

$\frac{1}{m} = - 4 m \pm 4 \sqrt{1 + m^{2}}$

$\Rightarrow 1 + 4 m^{2} = \pm 4 m \sqrt{1 + m^{2}} \Rightarrow 1 + 16 m^{4} + 8 m^{2} = 16 m^{2} + 16 m^{4}$

$\Rightarrow 8 m^{2} = 1 \Rightarrow m = \pm \frac{1}{2 \sqrt{2}}$

Point of contact of parabola is $(\frac{1}{{(\frac{1}{2 \sqrt{2}})}^{2}}, \frac{2}{\frac{1}{2 \sqrt{2}}})$ $[∵ Point of contact is (\frac{a}{m^{2}}, \frac{2 a}{m})]$

i.e., $(8, 4 \sqrt{2})$

Now, length of tangent $P Q = \sqrt{{(8 - 4)}^{2} + {(4 \sqrt{2})}^{2} - 16}$ $= \sqrt{32}$

$∴$ $P Q^{2} = 32$

Q 22 :

If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to _______. [2023]

(16)

Given, the $x$ -intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3.

$y^{2} = 8 x + 4 y + 4 \Rightarrow {(y - 2)}^{2} = 8 (x + 1) \Rightarrow Y^{2} = 4 \cdot 2 \cdot X$

$Y = y - 2, X = x + 1, a = 2$

For focus $x + 1 = 2$ and $y - 2 = 0 \Rightarrow x = 1$ and $y = 2$

$∴$ $Focus is (1, 2)$

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

$y - 2 = m (x - 1),$ $(3, 0)$ lies on the above line,

$∴$ $- 2 = m (3 - 1) \Rightarrow m = - 1$

$∴$ Equation of line is $y - 2 = - 1 (x - 1)$

i.e., $x + y - 3 = 0$ $∴$ $y = 3 - x$

Now put the value of $y$ in equation of parabola ${(3 - x)}^{2} = 8 x + 4 (3 - x) + 4$

$\Rightarrow 9 + x^{2} - 6 x = 8 x + 12 - 4 x + 4$ $∴$ $x = 5 + 4 \sqrt{2}, 5 - 4 \sqrt{2}$

So, $y = - 2 - 4 \sqrt{2},$ $4 \sqrt{2} - 2$

So, length of focal chord is

$= \sqrt{{(4 \sqrt{2} + 4 \sqrt{2})}^{2} + {(- 2 - 4 \sqrt{2} - 4 \sqrt{2} + 2)}^{2}}$

$= \sqrt{128 + 128} = \sqrt{256} = 16$

Q 23 :

Let S be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P (b, c), b, c \in N,$ on the parabola $y^{2} = 2 a x$ and the lines $x = b,$ $y = 0$ is 16 ${unit}^{2}$ , then $\sum_{a \in S} a$ is equal to _________ . [2023]

(146)

As $P (b, c)$ lies on parabola

So, $c^{2} = 2 a b$ ...(i)

Now, equation of tangent to parabola $y^{2} = 2 a x$ in point form is $y y_{1} = 2 a (\frac{x + x_{1}}{2}), where (x_{1}, y_{1}) \equiv (b, c)$

$\Rightarrow y c = a (x + b)$

For point $B$ , put $y = 0$ , we get $x = - b$

So, area of $∆ P B A$ , $\frac{1}{2} \times A B \times A P = 16 (Given)$

$\Rightarrow \frac{1}{2} \times 2 b \times c = 16 \Rightarrow b c = 16$

As $b$ and $c$ are natural numbers, possible values of $(b, c)$ are $(1, 16), (2, 8), (4, 4), (8, 2), (16, 1)$

Now from equation (i) $a = \frac{c^{2}}{2 b}$ and $a \in N,$ So value of $(b, c)$ are $(1, 16), (2, 8)$ and $(4, 4)$ . Now, values of $a$ are $128, 16$ and $2 .$

Hence, sum of values of $a$ is $146$ .

Q 24 :

Let O be the vertex of the parabola $x^{2}$ = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is: [2026]

$4 x - 5 y + 6 = 0$
$x - 2 y + 3 = 0$
$5 x - 4 y + 3 = 0$
$5 x - y - 3 = 0$

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(3)

$h = \frac{4 t}{5}$

$k = \frac{2 t^{2}}{5} = \frac{2}{5} {(\frac{5 h}{4})}^{2}$

$8 k = 5 h^{2}$

$\Rightarrow 5 x^{2} = 8 y$

$T = S_{1}$

$5 (x x_{1}) - 4 (y + y_{1}) = 5 x_{1}^{2} - 8 y_{1}$

$5 x - 4 (y + 2) = 5 - 8 \cdot 2$

$5 x - 4 y + 3 = 0$

Q 25 :

If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal to [2026]

284
292
280
288

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(4)

$(x_{1} y_{1}) = (3 t_{1}^{2}, 6 t_{1}) and (x_{2} y_{2}) = (3 t_{2}^{2}, 6 t_{2})$

$t_{1} t_{2} = - 4$

$x_{1} x_{2} = 9 {(t_{1} t_{2})}^{2}, y_{1} y_{2} = 36 t_{1} t_{2}$

$x_{1} x_{2} - y_{1} y_{2} = 9 (16) - 36 (- 4)$

$= 144 + 144$

$= 288$

Q 26 :

Let A be the focus of the parabola $y^{2} = 8 x .$ Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then ${(B C)}^{2}$ is equal to : [2026]

41
32
80
89

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(3)

$Coordinates of centroid of triangle ABC are$

$\frac{2}{3} (t_{1}^{2} + t_{2}^{2} + 1) = \frac{7}{3} \Rightarrow t_{1}^{2} + t_{2}^{2} = \frac{5}{2}$

$\frac{4}{3} (t_{1} + t_{2}) = \frac{4}{3} \Rightarrow t_{1} + t_{2} = 1$

${(t_{1} + t_{2})}^{2} = t_{1}^{2} + t_{2}^{2} + 2 t_{1} t_{2} \Rightarrow t_{1} t_{2} = - \frac{3}{4}$

${(t_{1} - t_{2})}^{2} = {(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2} = 4$

${(B C)}^{2} = 4 {(t_{1}^{2} - t_{2}^{2})}^{2} + 16 {(t_{1} - t_{2})}^{2}$

$\Rightarrow {(B C)}^{2} = 80$

Q 27 :

Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]

4
12
8
6

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(4)

$Parametric point P on x^{2} = 4 y is P (2 t, t^{2})$

$∴$ $mirror image of P in x - y = 1 is$

$Q \equiv (2 t - \frac{2.1 . (2 t - t^{2} - 1)}{2}, t^{2} + \frac{2.2 (1) . (2 t - t^{2} - 1)}{2})$

$Q \equiv (t^{2} + 1, 2 t - 1) \equiv (h, k)$

$∴$ $locus of Q is x = \frac{{(y + 1)}^{2}}{4} + 1$ which is the required parabola.

$∴$ ${(y + 1)}^{2} = 4 (x - 1)$

$∴$ $a = 1, b = 4, c = 1$

$∴$ $a + b + c = 6$

Q 28 :

An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is [2026]

$4 (6 + \sqrt{3})$
$4 (3 - \sqrt{3})$
$2 (3 + \sqrt{3})$
$2 (8 - 3 \sqrt{3})$

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(2)

$M_{O A} = \frac{2 t - 0}{t^{2} - 0} = \frac{2}{t}$

$\frac{2}{t} = \tan 30 °$

$t = 2 \sqrt{3}$

$Req. Circle: {(x - 12)}^{2} + y^{2} = {(4 \sqrt{3})}^{2}$

$Least distance = | C P - R |$

$= | 2 - 4 \sqrt{3} | = 4 (3 - \sqrt{3})$

Q 29 :

Let one end of a focal chord of the parabola $y^{2} = 16 x$ be (16,16). If P $(α, β)$ divides this focal chord internally in the ratio 5:2, then the minimum value of $α + β$ is equal to : [2026]

7
5
16
22

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(1)

$y^{2} = 16 x$

$∵$ $parameter of point A is t = 2$

$\Rightarrow Parameter of point B is t = - \frac{1}{2}$

$\Rightarrow Coordinates of B is (1, - 4)$

$Case 1:$

$α = \frac{5 + 32}{7} = \frac{37}{7}$

$β = \frac{- 20 + 32}{7} = \frac{12}{7}$

$\Rightarrow α + β = 7$

$Case 2:$

$α = \frac{2 + 80}{7}, β = \frac{- 8 + 80}{7}$

$α + β = 22$

$So minimum value of α + β = 7$

Q 30 :

Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 90 °$ . Then the locus of the centroid of such triangles OPA is : [2026]

$y^{2} - 4 x + 8 = 0$
$y^{2} - 6 x + 4 = 0$
$y^{2} - 2 x + 8 = 0$
$y^{2} - 9 x + 6 = 0$

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(3)

$m_{A P} = \frac{- t}{2}$

Equation of AP is

$y - 6 t = \frac{- t}{2} (x - 3 t^{2})$

Put $y = 0$

$\Rightarrow x = 12 + 3 t^{2}$

$\Rightarrow A (12 + 3 t^{2}, 0)$

Let centroid of $∆ O P A$ be $G (h, k)$

$\Rightarrow 3 h = 0 + 3 t^{2} + 12 + 3 t^{2}$

$3 k = 0 + 6 t + 0$

$\Rightarrow t = \frac{k}{2}, h = 2 t^{2} + 4$

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

$\Rightarrow Locus of (h, k) is$

$y^{2} = 2 x - 8$

Q 31 :

Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

$3 x^{2} = 2 y$
$2 y^{2} = 3 x$
$2 x^{2} = 3 y$
$3 y^{2} = 2 x$

1

2

3

4

(2)

$y^{2} = 4 x$

$Locus of mid point of O P$

$M (h, k) \Rightarrow h = \frac{t^{2}}{2}, k = t$

$\Rightarrow k^{2} = 2 h \Rightarrow y^{2} = 2 x$

$S : y^{2} = 2 x$

$\Rightarrow h = \frac{\frac{3 t^{2}}{2}}{4}, k = \frac{3 t}{4}$

$t^{2} = \frac{8 h}{3}, t = \frac{4 k}{3}$

$\Rightarrow \frac{16 k^{2}}{9} = \frac{8 h}{3}$ $\Rightarrow 2 k^{2} = 3 h$

$Locus of R : 2 y^{2} = 3 x$

Topic Question Set

Q 11 :

Let a tangent to the curve $y^{2} = 24 x$ meet the curve $x y = 2$ at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

Q 13 :

The equations of two sides of a variable triangle are $x = 0$ and $y = 3,$ and its third side is a tangent to the parabola $y^{2} = 6 x$ . The locus of its circumcentre is [2023]

Q 14 :

If the tangent at a point P on the parabola $y^{2} = 3 x$ is parallel to the line $x + 2 y = 1$ and the tangents at the points Q and R on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$ are perpendicular to the line $x - y = 2$ , then the area of the triangle PQR is [2023]

Q 15 :

If $P (h, k)$ be a point on the parabola $x = 4 y^{2}$ , which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola $y^{2} = 4 (x + y)$ is equal to [2023]

Q 16 :

The parabolas : $a x^{2} + 2 b x + c y = 0$ and $d x^{2} + 2 e x + f y = 0$ intersect on the line $y = 1$ . If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then [2023]

Q 17 :

Let A be a point on the x-axis. Common tangents are drawn from A to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at Q and R, then ${(Q R)}^{2}$ is equal to [2023]

Q 18 :

Let $y = f (x)$ represent a parabola with focus $(- \frac{1}{2}, 0)$ and directrix $y = - \frac{1}{2}$ . Then $S = {x \in ℝ : \tan^{- 1} (\sqrt{f (x)}) + \sin^{- 1} (\sqrt{f (x) + 1}) = \frac{π}{2}} :$ [2023]

Q 20 :

The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix $x = - 3$ are in the ratio 3 : 1. If $R (α, β)$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{β^{2}}{α}$ is equal to _______. [2023]

Q 21 :

Let a common tangent to the curves $y^{2} = 4 x$ and ${(x - 4)}^{2} + y^{2} = 16$ touch the curves at the points P and Q. Then ${(P Q)}^{2}$ is equal to _________ . [2023]

Q 22 :

If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to _______. [2023]

Q 23 :

Let S be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P (b, c), b, c \in N,$ on the parabola $y^{2} = 2 a x$ and the lines $x = b,$ $y = 0$ is 16 ${unit}^{2}$ , then $\sum_{a \in S} a$ is equal to _________ . [2023]

Q 24 :

Let O be the vertex of the parabola $x^{2}$ = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is: [2026]

Q 25 :

If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal to [2026]

Q 26 :

Let A be the focus of the parabola $y^{2} = 8 x .$ Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then ${(B C)}^{2}$ is equal to : [2026]

Q 27 :

Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]

Q 28 :

An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is [2026]

Q 29 :

Let one end of a focal chord of the parabola $y^{2} = 16 x$ be (16,16). If P $(α, β)$ divides this focal chord internally in the ratio 5:2, then the minimum value of $α + β$ is equal to : [2026]

Q 30 :

Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 90 °$ . Then the locus of the centroid of such triangles OPA is : [2026]

Q 31 :

Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

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Topic Question Set

Q 11 : Let a tangent to the curve y2=24x meet the curve xy=2 at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

Q 12 : The equations of the sides AB and AC of a triangle ABC are (λ+1)x+λy=4 and λx+(1-λ)y+λ=0 respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the be part of the parabola y2=6x in the first quadrant is [2023]

Q 13 : The equations of two sides of a variable triangle are x=0 and y=3, and its third side is a tangent to the parabola y2=6x. The locus of its circumcentre is [2023]

Q 14 : If the tangent at a point P on the parabola y2=3x is parallel to the line x+2y=1and the tangents at the points Q and R on the ellipse x24+y21=1 are perpendicular to the line x-y=2, then the area of the triangle PQR is [2023]

Q 15 : If P(h,k) be a point on the parabola x=4y2, which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola y2=4(x+y) is equal to [2023]

Q 16 : The parabolas : ax2+2bx+cy=0 and dx2+2ex+fy=0 intersect on the line y=1. If a,b,c,d,e,f are positive real numbers and a,b,c are in G.P., then [2023]

Q 17 : Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2+y2=8 and y2=16x. If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to [2023]

Q 18 : Let y=f(x) represent a parabola with focus (-12,0) and directrix y=-12. Then S={x∈ℝ: tan-1(f(x))+sin-1(f(x)+1)=π2}: [2023]

Q 19 : Let the tangent to the curve x2+2x-4y+9=0 at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x-3y=6 meet the parabola y2=4x at B. If B lies on the line 2x-3y=8, then (AB)2 is equal to ________ . [2023]

Q 20 : The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix x=-3 are in the ratio 3 : 1. If R(α,β) is the point of intersection of the tangents to the parabola at P and Q, then β2α is equal to _______. [2023]

Q 21 : Let a common tangent to the curves y2=4x and (x-4)2+y2=16 touch the curves at the points P and Q. Then (PQ)2 is equal to _________ . [2023]

Q 22 : If the x-intercept of a focal chord of the parabola y2=8x+4y+4 is 3, then the length of this chord is equal to _______. [2023]

Q 23 : Let S be the set of all a∈N such that the area of the triangle formed by the tangent at the point P(b,c), b,c∈N, on the parabola y2=2ax and the lines x=b,y=0 is 16 unit2, then ∑a∈Sa is equal to _________ . [2023]

Q 24 : Let O be the vertex of the parabola x2 = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is: [2026]

Q 25 : If the chord joining the points P1(x1,y1) and P2(x2,y2) on the parabola y2=12x subtends a right angle at the vertex of the parabola, then x1x2-y1y2 is equal to [2026]

Q 26 : Let A be the focus of the parabola y2=8x. Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is (73,43), then (BC)2 is equal to : [2026]

Q 27 : Let the image of parabola x2=4y, in the line x-y=1 be (y+a)2=b(x-c), a,b,c∈ℕ. Then a+b+c is equal to [2026]

Q 28 : An equilateral triangle OAB is inscribed in the parabola y2=4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is [2026]

Q 29 : Let one end of a focal chord of the parabola y2=16x be (16,16). If P(α,β) divides this focal chord internally in the ratio 5:2, then the minimum value of α+β is equal to : [2026]

Q 30 : Let y2=12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that ∠OPA=90°. Then the locus of the centroid of such triangles OPA is : [2026]

Q 31 : Let the locus of the mid-point of the chord through the origin O of the parabola y2=4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

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Q 11 :

Let a tangent to the curve $y^{2} = 24 x$ meet the curve $x y = 2$ at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

Q 13 :

The equations of two sides of a variable triangle are $x = 0$ and $y = 3,$ and its third side is a tangent to the parabola $y^{2} = 6 x$ . The locus of its circumcentre is [2023]

Q 14 :

If the tangent at a point P on the parabola $y^{2} = 3 x$ is parallel to the line $x + 2 y = 1$ and the tangents at the points Q and R on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$ are perpendicular to the line $x - y = 2$ , then the area of the triangle PQR is [2023]

Q 15 :

If $P (h, k)$ be a point on the parabola $x = 4 y^{2}$ , which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola $y^{2} = 4 (x + y)$ is equal to [2023]

Q 16 :

The parabolas : $a x^{2} + 2 b x + c y = 0$ and $d x^{2} + 2 e x + f y = 0$ intersect on the line $y = 1$ . If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then [2023]

Q 17 :

Let A be a point on the x-axis. Common tangents are drawn from A to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at Q and R, then ${(Q R)}^{2}$ is equal to [2023]

Q 18 :

Let $y = f (x)$ represent a parabola with focus $(- \frac{1}{2}, 0)$ and directrix $y = - \frac{1}{2}$ . Then $S = {x \in ℝ : \tan^{- 1} (\sqrt{f (x)}) + \sin^{- 1} (\sqrt{f (x) + 1}) = \frac{π}{2}} :$ [2023]

Q 20 :

The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix $x = - 3$ are in the ratio 3 : 1. If $R (α, β)$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{β^{2}}{α}$ is equal to _______. [2023]

Q 21 :

Let a common tangent to the curves $y^{2} = 4 x$ and ${(x - 4)}^{2} + y^{2} = 16$ touch the curves at the points P and Q. Then ${(P Q)}^{2}$ is equal to _________ . [2023]

Q 22 :

If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to _______. [2023]

Q 23 :

Let S be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P (b, c), b, c \in N,$ on the parabola $y^{2} = 2 a x$ and the lines $x = b,$ $y = 0$ is 16 ${unit}^{2}$ , then $\sum_{a \in S} a$ is equal to _________ . [2023]

Q 24 :

Let O be the vertex of the parabola $x^{2}$ = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is: [2026]

Q 25 :

If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal to [2026]

Q 26 :

Let A be the focus of the parabola $y^{2} = 8 x .$ Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then ${(B C)}^{2}$ is equal to : [2026]

Q 27 :

Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]

Q 28 :

An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is [2026]

Q 29 :

Let one end of a focal chord of the parabola $y^{2} = 16 x$ be (16,16). If P $(α, β)$ divides this focal chord internally in the ratio 5:2, then the minimum value of $α + β$ is equal to : [2026]

Q 30 :

Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 90 °$ . Then the locus of the centroid of such triangles OPA is : [2026]

Q 31 :

Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]