JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the foci and length of the latus rectum of an ellipse \(\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) be \(( \pm 5,0)\) and \(\sqrt{50}\), respectively. Then, the square of the eccentricity of the hyperbola \(\frac{\mathrm{x}^2}{\mathrm{~b}^2}-\frac{\mathrm{y}^2}{\mathrm{a}^2 \mathrm{~b}^2}=1\) equals
- A \(40\)
- B \(48\)
- C \(51\)
- D \(50\)
Answer & Solution
Correct Answer
(C) \(51\)
Step-by-step Solution
Detailed explanation
\(\text { focii } \equiv( \pm 5,0) ; \frac{2 b^2}{a}=\sqrt{50} \) \(ae=5 \quad b^2=\frac{5 \sqrt{2} a}{2} \) \(b^2=a^2\left(1-e^2\right)=\frac{5 \sqrt{2} a}{2}\) \( \Rightarrow \mathrm{a}\left(1-\mathrm{e}^2\right)=\frac{5 \sqrt{2}}{2}\)…
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 three vectors \(\vec{a}, \overrightarrow{\mathrm{b}}\) and \(\vec{c}\) be such that \(\vec{a} \times \overrightarrow{\mathrm{b}}=\vec{c}, \overrightarrow{\mathrm{b}} \times \vec{c}=\vec{a}\) and \(|\vec{a}|=2\) Then which one of the following is not true?JEE Mains 2021 Medium
- Let \(A=\left\{(x, y) \in R ^2: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^2}\right\}\) and \(B=\left\{(x, y) \in R \times R : 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^2}\right\}\right\}\) Then the ratio of the area of \(A\) to the area of \(B\) isJEE Mains 2023 Hard
- Let \(\left\{a_{n}\right\}_{n-1}^{\infty}\) be a sequence such that \(a_{1}=1, a_{2}=1\) and \(a_{n+2}=2 a_{n+1}+a_{n}\) for all \(n \geq 1 .\) Then tha value of \(47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3 n}}\) is equal to \(.....\)JEE Mains 2021 Hard
- A function \(f(x)\) is given by \(f(x)=\frac{5^{x}}{5^{x}+5}\), then the sum of the series \(f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)\) is equal to ....... .JEE Mains 2021 Medium
- The sum of the first three terms of a \(G.P.\) is \(S\) and their product is \(27 .\) Then all such \(S\) lie inJEE Mains 2020 Medium
- The sum of all rational terms in the expansion of \((2+\sqrt{3})^8\) isJEE Mains 2025 Easy
More PYQs from JEE Mains
- Let \(A , B , C\) be three points whose position vectors respectively are: \(\overrightarrow{ a }=\hat{ i }+4 \hat{ j }+3 \hat{ k }\) ; \(\overrightarrow{ b }=2 \hat{ i }+\alpha \hat{ j }+4 \hat{ k }, \alpha \in R\) ;\(\overrightarrow{ c }=3 \hat{ i }-2 \hat{ j }+5 \hat{ k }\) . If \(\alpha\) is the smallest positive integer for which \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear, then the length of the median, in \(\triangle ABC\), through \(A\) isJEE Mains 2022 Easy
- Let \(x=2\) be a root of the equation \(x^2+p x+q=0\) and \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos \left(x^2-4 p x+q^2+8 q+16\right)}{(x-2 p)^4}, & x \neq 2 p \\ 0, & x=2 p\end{array}\right.\) Then \(\lim _{x \rightarrow 22^{+}}[f(x)]\) where [. ] denotes greatest integer function, is \(........\)JEE Mains 2023 Hard
- Let \(\lambda_1, \lambda_2\) be the values of \(\lambda\) for which the points \(\left(\frac{5}{2}, 1, \lambda\right)\) and \((-2,0,1)\) are at equal distance from the plane \(2 x+3 y-6 z+7=0\). if \(\lambda_1 > \lambda_2\), then the distance of the point \(\left(\lambda_1-\lambda_2, \lambda_2, \lambda_1\right)\) from the line \(\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}\) is \(............\).JEE Mains 2023 Hard
- \(\int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x\) is equal toJEE Mains 2023 Hard
- If \(\alpha ,\beta \ne 0\) and \(f\left( n \right) = {\alpha ^n} + {\beta ^n}\) and \(\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 - \alpha } \right)^2}\) \({\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}\) ,then \(K=\) . . . . . .JEE Mains 2014 Hard
- Let \(f:(-2,2) \rightarrow\) IR be defined by \(f(x)=\left\{\begin{array}{cc}x[x] & ,-2 < x < 0 \\(x-1)[x] & , 0 \leq x < 2\end{array}\right.\) Where \([x]\) denotes the greatest integer function. If \(m\) and \(n\) respectively are the number of points in \((-2,2)\) at which \(y =|f(x)|\) is not continuous and not differentiable, then \(m + n\) is equal to \(...........\).JEE Mains 2023 Hard