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

Q 31 :
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 32 :
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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$A B : (λ + 1) x + λ y = 4$

$A C : λ x + (1 - λ) y + λ = 0$

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 33 :
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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$Equation of the parabola is y^{2} = 6 x .$

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 34 :
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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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 35 :
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
2
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 36 :
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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$a x^{2} + 2 b x + c y = 0$ ...(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 37 :
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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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 38 :
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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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 39 :
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]

0%

(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 40 :
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]

0%

(16)

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

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

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

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

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

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

Topic Question Set

Q 31 :
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 33 :
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 34 :
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 35 :
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 36 :
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 37 :
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 38 :
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]

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Q 40 :
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]

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Topic Question Set

Q 31 : 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 32 : 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 33 : 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 34 : 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 35 : 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 36 : 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 37 : 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 38 : 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 39 : 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]

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Q 40 : 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]

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Q 31 :
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 33 :
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 34 :
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 35 :
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 36 :
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 37 :
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 38 :
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 40 :
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]