WBJEE · Maths · Hyperbola
A double ordinate \(\mathrm{PQ}\) of the hyperbola \(\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\) is such that \(\Delta \mathrm{OPQ}\) is equilateral, \(\mathrm{O}\) being the centre of the hyperbola. Then the eccentricity e satisfies the relation
- A \(1 < \mathrm{e} < \frac{2}{\sqrt{3}}\)
- B \(\mathrm{e}=\frac{2}{\sqrt{3}}\)
- C \(\mathrm{e}=\frac{\sqrt{3}}{2}\)
- D \(\mathrm{e}>\frac{2}{\sqrt{3}}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{e}>\frac{2}{\sqrt{3}}\)
Step-by-step Solution
Detailed explanation
Hint: \(P \equiv(a \sec \theta, b \tan \theta) \angle \mathrm{POM}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{\mathrm{b}}{\mathrm{a}} \sin \theta\) \(\Rightarrow \sin \theta=\frac{a}{b \sqrt{3}}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- For each \(n \in N, 2^{3 n}-1\) is divisible by
where \(\mathrm{N}\) is a set of natural numbersWBJEE 2009 Hard - If \(\frac{1}{{ }^{5} C_{r}}+\frac{1}{{ }^{6} C_{r}}=\frac{1}{{ }^{4} C_{r}},\) then the value of \(r\) isWBJEE 2016 Medium
- The number of real roots of equation \(log_ex+e x=0\) isWBJEE 2015 Easy
- The value of the integral \(\int_{1}^{5}[|x-3|+|1-x|] d x\) is equal toWBJEE 2012 Hard
- If \(\mathrm{a}=2 \sqrt{2}, \mathrm{~b}=6, \mathrm{~A}=45^{\circ}\), thenWBJEE 2009 Medium
- The value of \(\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]\) is equal toWBJEE 2019 Easy
More PYQs from WBJEE
- \(A, B\) are fixed points with coordinates \((0, a)\) and \((0, b)(a > 0, b > 0)\). \(P\) is variable point \((x, 0)\) referred to rectangular axis. If the angle \(\angle \mathrm{APB}\) is maximum, thenWBJEE 2023 Medium
- The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e isWBJEE 2019 Easy
- A wire of resistance \(5 \Omega\) is drawn out so that its new length is 3 times its original length. What is the reistance of the new wire?WBJEE 2009 Medium
- Let \(p, q\) and \(r\) be the side4s opposite to the angles \(P, Q\) and \(R,\) respectively in a \(\Delta P Q R\). Then, \(2 p r \sin \left(\frac{p-Q+R}{2}\right)\) equalsWBJEE 2012 Easy
- The height of a waterfall is \(50 \mathrm{~m}\). If \(g=9.8 \mathrm{~ms}^{-2}\) the difference between the temperature at the top and the bottom of the waterfall is:WBJEE 2009 Hard
- Let \(S\) be the set of points, whose abscissae and ordinates are natural numbers. Let \(P \in S\), such that the sum of the distance of \(P\) from (8,0) and (0,12) is minimum among all elements in \(S .\) Then, the number of such points \(P\) in \(S\) isWBJEE 2016 Medium