Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • A tangent to the hyperbola <img alt="\frac{x^{2}}{4}-\frac{y^{2}}{2}=1" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cfrac%7By%5E%7B2%7D%7D%7B2%7D%3D1

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

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

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


Answers (1)

best_answer

\begin{aligned} &\text { Equation of the tangent at the point ' } \theta \text { ' is }\\ &\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 \end{aligned}

\\\Rightarrow \quad \frac{k}{h}=\frac{-b}{a \sin \theta} \Rightarrow \sin \theta=\frac{-b h}{a k} \\ \cos \theta=\frac{h}{a}

square and add, we get

\\\frac{b^{2} h^{2}}{a^{2} k^{2}}+\frac{h^{2}}{a^{2}}=1\\ \Rightarrow \frac{b^{2}}{k^{2}}+1=\frac{a^{2}}{h^{2}} \\ \Rightarrow \frac{a^{2}}{h^{2}}-\frac{b^{2}}{k^{2}}=1

\begin{aligned} &\text { Now, given eq }^{n} \text { of hyperbola is } \frac{x^{2}}{4}-\frac{y^{2}}{2}=1\\ &\Rightarrow a^{2}=4, b^{2}=2\\ &\therefore \mathrm{R} \text { lies on } \frac{a^{2}}{x^{2}}-\frac{b^{2}}{y^{2}}=1 \text { i.e., } \frac{4}{x^{2}}-\frac{2}{y^{2}}=1 \end{aligned}

Posted by

Info Expert 30

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 Paper 2 Toppers: 4 score 100 percentile; most from Kerala, one female topper
anam-khan

JEE Mains 2026 paper 2 result out on jeemain.nta.nic.in
anam-khan

JEE Mains 2026 session 2 B.Arch, B.Planning final answer key out, no question dropped

Latest Question