JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the sum of the focal distances of the point \(\mathrm{P}(4,3)\) on the hyperbola \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1\) be \(8 \sqrt{\frac{5}{3}}\). If for \(H\), the length of the latus rectum is \(l\) and the product of the focal distances of the point P is m , then \(9 l^2+6 \mathrm{~m}\) is equal to :-
- A \(184\)
- B \(186\)
- C \(185\)
- D \(187\)
Answer & Solution
Correct Answer
(C) \(185\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & e x+a+e x-a=8 \sqrt{\frac{5}{3}} \\ & 2 e x=8 \sqrt{\frac{5}{3}} \\ & 2 e \times 4=8 \sqrt{\frac{5}{3}} \\ & e=\sqrt{\frac{5}{3}} \\ & b^2=a^2\left(\left(\frac{\sqrt{5}}{3}\right)^2-1\right) \\ & b^2=\frac{2}{3} a^2\end{aligned}\)…
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
- The letters of the word \(OUGHT\) are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word \(TOUGH\) is :JEE Mains 2023 Hard
- If \(x\,{\log _e}({\log _e}\,\,x)\, - \,{x^2} + {y^2} = 4\,(y\, > \,0),\) then \(\frac{{dy}}{{dx}}\) at \(x = e\) is equal toJEE Mains 2019 Hard
- If the variance of the terms in an increasing \(A.P.\), \(b _{1}, b _{2}, b _{3}, \ldots b _{11}\) is \(90,\) then the common difference of this \(A.P.\) isJEE Mains 2020 Medium
- \(50^{\text {th }}\) root of a number \(x\) is \(12\) and \(50^{\text {th }}\) root of another number \(y\) is \(18\) . Then the remainder obtained on dividing \(( x + y )\) by \(25\) is \(........\).JEE Mains 2023 Hard
- If \((20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\ldots \ldots\). \(+20(21)^{19}= k (20)^{19}\), then \(k\) is equal toJEE Mains 2023 Hard
- Let \(l_{1}\) be the line in \(xy\)-plane with \(x\) and \(y\) intercepts \(\frac{1}{8}\) and \(\frac{1}{4 \sqrt{2}}\) respectively, and \(l_{2}\) be the line in \(zx\)-plane with \(x\) and \(z\) intercepts \(-\frac{1}{8}\) and \(-\frac{1}{6 \sqrt{3}}\) respectively. If \(d\) is the shortest distance between the line \(l_{1}\) and \(l_{2}\), then \(d ^{-2}\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) isJEE Mains 2019 Medium
- Let \(\mathrm{m}\) and \(\mathrm{n}\) be the coefficients of seventh and thirteenth terms respectively in the expansion of \(\left(\frac{1}{3} \mathrm{x}^{\frac{1}{3}}+\frac{1}{2 \mathrm{x}^{\frac{2}{3}}}\right)^{18}\). Then \(\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}\) is :JEE Mains 2024 Hard
- Let \(f: [1, \infty) \rightarrow \mathbf{R}\) be a differentiable function defined as \(f(x) = \int_1^x f(t)\,dt + (1-x)(\log_e x - 1) + e\). Then the value of \(f(f(1))\) is :JEE Mains 2026 Hard
- The number of solutions of the equation \(2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}\right\}\) isJEE Mains 2025 Easy
- Let \([ t ]\) denote the greatest integer \(\leq t\) and \(\{ t \}\) denote the fractional part of \(t\). Then integral value of \(\alpha\) for which the left hand limit of the function \(f(x)=[1+x]+\frac{\alpha^{2[x]+[x]}+[x]-1}{2[x]+\{x\}}\) at \(x=0\) is equal to \(\alpha-\frac{4}{3}\) isJEE Mains 2022 Medium
- If \(f:R \to R\) is a differentiable function and \(f\left( 2 \right) = 6\), then \(\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {\frac{{2\,tdt}}{{\left( {x - 2} \right)}}} \) isJEE Mains 2019 Hard