JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let PQ be a chord of the hyperbola \( \frac{x^{2}}{4}-\frac{y^{2}}{b^{2}}=1 \), perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is \( \sqrt{3} \) then the area of the triangle OPQ is:
- A \( 2\sqrt{3} \)
- B \( \frac{8\sqrt{3}}{5} \)
- C \( \frac{11}{5} \)
- D \( \frac{9}{5} \)
Answer & Solution
Correct Answer
(B) \( \frac{8\sqrt{3}}{5} \)
Step-by-step Solution
Detailed explanation
\(e=\sqrt{1+\frac{b}{4}}=\sqrt{3}\) \(\Rightarrow b =8\) ∴ Hyperbola \(\frac{ x ^2}{4}-\frac{ y ^2}{8}=1\) \(\frac{ PM }{ OM }=\tan 30^{\circ}\) \(\Rightarrow \frac{2 \sqrt{2} \tan \theta}{2 \sec \theta}=\frac{1}{\sqrt{3}} \Rightarrow \sin \theta=\frac{1}{\sqrt{6}}\) Area…
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
- Let \(\mathrm{R}=\{(1,2),(2,3),(3,3)\}\) be a relation defined on the set \(\{1,2,3,4\}\). Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:JEE Mains 2025 Easy
- If the area of the region {(x, y) : \( 1-2x\le y\le4-x^{2}, x\ge0,y\ge0 \)} is \( \frac{\alpha}{\beta}, \alpha, \beta \in N \), gcd(\(α,β\))=1, then the value of \( (\alpha+\beta) \) is :JEE Mains 2026 Easy
- Consider three observations \(a, b\) and \(c\) such that \(b = a + c .\) If the standard deviation of \(a +2\) \(b +2, c +2\) is \(d ,\) then which of the following is true ?JEE Mains 2021 Medium
- Let the position vectors of the points \(A , B , C\) and \(D\) be \(5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(-\hat{ i }+5 \hat{ j }+6 \hat{ k }\). Let the set \(S =\{\lambda \in R\) : The points \(A\), \(B , C\) and D are coplanar \(\}\). Then \(\sum_{\lambda \in S}(\lambda+2)^2\) is equal toJEE Mains 2023 Hard
- Let \(A , B , C\) be \(3 \times 3\) matrices such that \(A\) is symmetric and \(B\) and \(C\) are skew-symmetric.Consider the statements \((S1): A ^{13} B ^{26}- B ^{26} A ^{13}\) is symmetric \((S2):A ^{26} C ^{13}- C ^{13} A ^{26}\) is symmetric Then,JEE Mains 2023 Hard
- Let a vertical tower \(AB\) have its end \(A\) on the level ground. Let \(C\) be the mid-point of \(AB\) and \(P\) be apoint on the ground such that \(AP=2AB\) . If \(\angle BPC = \beta \) then \(\tan \beta \) is equal to :JEE Mains 2017 Hard
More PYQs from JEE Mains
- Let \(a \in R\) and let \(\alpha, \beta\) be the roots of the equation \(x^2+60^{\frac{1}{4}} x+a=0\). If \(\alpha^4+\beta^4=-30\), then the product of all possible values of \(a\) is \(......\)JEE Mains 2023 Hard
- Let the curve \(z(1+i)+\bar{z}(1-i)=4, z \in \mathrm{C}\), divide the region \(|z-3| \leq 1\) into two parts of areas \(\alpha\) and \(\beta\). Then \(|\alpha-\beta|\) equals :JEE Mains 2025 Medium
- The function \(f:(-\infty, \infty) \rightarrow(-\infty, 1),\) defined by \(f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}} \text { is : }\)JEE Mains 2025 Easy
- If the function \(f\) defined on \(\left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) by \(f\,(x)\, = \,\left\{ {\begin{array}{*{20}{c}}
{\frac{{\sqrt 2 \,\cos \,x - \,1}}{{\cot \,x\, - \,1}}\,,\,x\, \ne \,\frac{\pi }{4}}\\
{k,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \frac{\pi }{4}}
\end{array}} \right.\) is continuous, then \(k\) is equal toJEE Mains 2019 Hard - A player \(X\) has a biased coin whose probability of showing heads is \(p\) and a player \(Y\) has a fair coin . They start playing a game with their own coins and play alternately . The player who throws a head first is a winner. If \(X\) starts the game, and the probability of winning the game by both the players is equal, then the value of \('p'\) isJEE Mains 2018 Hard
- The equation \( lm\,\left( {\frac{{iz - 2}}{{z - i}}} \right) + 1 = 0\,,z \in C\,,z \ne i\) represents a part of a circle having radius equal toJEE Mains 2017 Hard