A tangent to the hyperbola (x^2)/(4)-(y^2)/(2)=1 meets x -axis at P and y -axis at Q . Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). Then R lies on :

Correct answer: D. (4)/(x^2)-(2)/(y^2)=1 Equation of the tangent at the point ' θ ' is aligned & (x θ)/(a)-(y θ)/(b)=1 & P=(a θ,0) and Q=(0,-b θ) & Let R be (h, k) h=a θ, k=-b θ & (k)/(h)=(-b)/(a θ) θ=(-b h)/(a k) and & θ=(h)/(a) aligned By squaring and adding, (b^2 h^2)/(a^2 k^2)+(h^2)/(a^2)= 1 aligned & (b^2)/(k^2)+1=(a^2)/(h^2) & (a^2)/(h^2)-(b^2)/(k^2)=1 aligned Now, given ^ n of hyperbola is (x^2)/(4)-(y^2)/(2)=1 a²=4, b²=2 R lies on (a^2)/(x^2)-(b^2)/(y^2)=1 i.e., (4)/(x^2)-(2)/(y^2)=1

JEE Main 201323 AprilOnline

Exam: JEE Main
Year: 2013
Date: 23 April
Mode: Online
Subject: Mathematics
Chapter: Hyperbola

JEE Main 2013 · 23 April · Online · Mathematics · Hyperbola

A tangent to the hyperbola (x^2)/(4)-(y^2)/(2)=1 meets x -axis at P...

A tangent to the hyperbola

\frac{x^2}{4}-\frac{y^2}{2}=1

meets

x

-axis at

\mathrm{P}

and

y

-axis at

\mathrm{Q}

. Lines

\mathrm{PR}

and

\mathrm{QR}

are drawn such that OPRQ is a rectangle (where

\mathrm{O}

is the origin). Then

\mathrm{R}

lies on :

A tangent to the hyperbola (x^2)/(4)-(y^2)/(2)=1 meets x -axis at P and y -axis at Q . Lines PR and QR are drawn such

Options

A $\frac{4}{x^2}+\frac{2}{y^2}=1$
B $\frac{2}{x^2}-\frac{4}{y^2}=1$
C $\frac{2}{x^2}+\frac{4}{y^2}=1$
D $\frac{4}{x^2}-\frac{2}{y^2}=1$

Correct answer

D. $\frac{4}{x^2}-\frac{2}{y^2}=1$

Step-by-step solution

Equation of the tangent at the point '

\theta

' is

\begin{aligned}& \frac{x \sec \theta}{a}-\frac{y \tan \theta}{b}=1 \\ & \Rightarrow P=(a \cos \theta,0) \text{ and }Q=(0,-b \cot\theta)\\ & \text{ Let } \mathrm{R} \text{ be }(\mathrm{h}, \mathrm{k}) \Rightarrow \mathrm{h}=a \cos \theta, \mathrm{k}=-b \cot \theta\\ & \Rightarrow \quad \frac{k}{h}=\frac{-b}{a \sin \theta} \Rightarrow \sin \theta=\frac{-b h}{a k} \quad \text{ and }\\ & \cos\theta=\frac{h}{a} \end{aligned}

By squaring and adding,

\frac{b^2 h^2}{a^2 k^2}+\frac{h^2}{a^2}=

<img src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fjee_main%252FJtEjEWsSw29B2ya_7VeKey3oSqECvSRPtP37PbU6aNk.original.fullsize.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2FJtEjEWsSw29B2ya_7VeKey3oSqECvSRPtP37PbU6aNk.original.fullsize.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2FJtEjEWsSw29B2ya_7VeKey3oSqECvSRPtP37PbU6aNk.original.fullsize.png?alt=media">

\begin{aligned}& \Rightarrow \frac{b^2}{k^2}+1=\frac{a^2}{h^2}\\ & \Rightarrow \frac{a^2}{h^2}-\frac{b^2}{k^2}=1 \end{aligned}

Now, given

\mathrmq^{\mathrm{n}}

of hyperbola is

\frac{x^2}{4}-\frac{y^2}{2}=1

\Rightarrow a²=4, b²=2

\therefore

R lies on

\frac{a^2}{x^2}-\frac{b^2}{y^2}=1

i.e., $\frac{4}{x^2}-\frac{2}{y^2}=1

Practice Hyperbola on Quantrex Academy →

More from Hyperbola

If the eccentricity e of the hyperbola (x^2)/(a^2) - (y^2)/(b^2) = 1 , passing through… JEE Main 2026 Let the eccentricity e of a hyperbola satisfy the equation 6e^2 - 11e + 3 = 0 . If the… JEE Main 2026 Let H: (x^2)/(a^2)-(y^2)/(b^2)=1 be a hyperbola such that the distance between its foci… JEE Main 2026 Let O be the origin, and P and Q be two points on the rectangular hyperbola xy = 12 such… JEE Main 2026 For some θ ∈(0, (π)/(2)) , let the eccentricity and the length of the latus rectum of the… JEE Main 2026 Let PQ be a chord of the hyperbola x^ 2 4 - y^ 2 b^ 2 =1 , perpendicular to the x -axis… JEE Main 2026 Let the domain of the function f(x)= _ 3 _ 5 _ 7 (9 x-x^ 2 -13) be the interval (m, n) .… JEE Main 2026 Let P(10,2 15 ) be a point on the hyperbola x^ 2 a^ 2 - y^ 2 ~b^ 2 =1 , whose foci are S… JEE Main 2026 If the line α x+2 y=1 , where α ∈ ℝ , does not meet the hyperbola x^ 2 -9 y^ 2 =9 , then… JEE Main 2026 Let the foci of a hyperbola coincide with the foci of the ellipse x^ 2 36 + y^ 2 16 =1 .… JEE Main 2026 Let e_1 and e_2 be the eccentricities of the ellipse x^2 ~b^2 + y^2 25 =1 and the… JEE Main 2025 Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be… JEE Main 2025 Consider the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 having one of its focus at P(-3,0) . If… JEE Main 2025 Let the sum of the focal distances of the point P(4,3) on the hyperbola H: x^2 a^2 - y^2… JEE Main 2025 If the equation of the hyperbola with foci (4,2) and (8,2) is 3 x^2-y^2-α x+β y+γ=0… JEE Main 2025 Let the product of the focal distances of the point P(4,2 3 ) on the hyperbola H: x^2 a^2… JEE Main 2025 Full Hyperbola list Hyperbola JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs