JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let \(H: \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) be a hyperbola such that the distance between its foci is \(6\) and the distance between its directrices is \(\dfrac{8}{3}\). If the line \(x=\alpha\) intersects the hyperbola \(H\) at the points \(A\) and \(B\) such that the area of the triangle \(AOB\) is \(4\sqrt{15}\), where \(O\) is the origin, then \(\alpha^2\) equals
- A \(12\)
- B \(16\)
- C \(24\)
- D \(25\)
Answer & Solution
Correct Answer
(B) \(16\)
Step-by-step Solution
Detailed explanation
Given distance between foci is \(2ae = 6 \Rightarrow ae = 3\) Distance between directrices is \(\dfrac{2a}{e} = \dfrac{8}{3}\) Multiplying the two equations, we get \(4a^2 = 16 \Rightarrow a^2 = 4\) Dividing the two equations, we get \(e^2 = \dfrac{9}{4}\) Using…
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 \(\left(1+x+2 x^{2}\right)^{20}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{40} x^{40}\) then \(a _{1}+ a _{3}+ a _{5}+\ldots+ a _{37}\) is equal toJEE Mains 2021 Hard
- Let \(P\) the point of intersection of the lines \(\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}\) and \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}\). Then, the shortest distance of \(\mathrm{P}\) from the line \(4 \mathrm{x}=2 \mathrm{y}=\mathrm{z}\) isJEE Mains 2024 Medium
- The function/ defined by \(f(x)\, = x^3 - 3x^2 + 5x + 7\), isJEE Mains 2017 Hard
- If \(\left\{a_{i}\right\}_{i=1}^{n}\) where \(n\) is an even integer, is an arithmetic progression with common difference \(1\) , and \(\sum \limits_{ i =1}^{ n } a _{ i }=192, \sum \limits_{ i =1}^{ n / 2} a _{2 i }=120\), then \(n\) is equal toJEE Mains 2022 Hard
- In a tournament, a team plays \(10\) matches with probabilities of winning and losing each match as \(\frac{1}{3}\) and \(\frac{2}{3}\) respectively. Let \(x\) be the number of matches that the team wins, and \(y\) be the number of matches that team loses. If the probability \(\mathrm{P}(|\mathrm{x}-\mathrm{y}| \leq 2)\) is \(\mathrm{p}\), then \(3^9 \mathrm{p}\) equals ...........JEE Mains 2024 Hard
- A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the work in 24 days less than mason B alone. Then mason A alone will complete the work in:JEE Mains 2026 Hard
More PYQs from JEE Mains
- If the sum of the deviations of \(50\) observations from \(30\) is \(50\), then the mean of these observations isJEE Mains 2019 Hard
- If some three consecutive in the binomial expansion of \({\left( {x + 1} \right)^n}\) in powers of \(x\) are in the ratio \(2 : 15 : 70\), then the average of these three coefficient isJEE Mains 2019 Hard
- Let \(K\) be the set of all real values of \(x\) where the function \(f\left( x \right) = \sin \,\left| x \right| - \left| x \right| + 2\,\left( {x - \pi } \right)\,\cos \,\left| x \right|\) is not differentiable. Then the set \(K\) is equal toJEE Mains 2019 Hard
- Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and R be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m} \mathrm{n}\) is equal to :-JEE Mains 2025 Easy
- For \(p, q \in R\), consider the real valued function \(f ( x )=( x - p )^{2}- q , x \in R\) and \(q >0\). Let \(a _{1}, a _{2}, a _{3}\) and \(a _{4}\) be in an arithmetic progression with mean \(P\) and positive common difference. If \(\left| f \left( a _{ i }\right)\right|=500\) for all \(i=1,2,3,4\), then the absolute difference between the roots of \(f ( x )=0\) is.JEE Mains 2022 Hard
- Let the mean and standard deviation of marks of class \(A\) of \(100\) students be respectively \(40\) and \(\alpha( > 0)\), and the mean and standard deviation of marks of class \(B\) of \(n\) students be respectively \(55\) and \(30-\alpha\). If the mean and variance of the marks of the combined class of \(100+ n\) students are respectively \(50\) and \(350\),then the sum of variances of classes \(A\) and \(B\) isJEE Mains 2023 Hard