JEE Main
- Exam
- JEE Main
- Subject
- Mathematics
- Chapter
- Hyperbola
JEE Main · Mathematics · Hyperbola
Let e_1 be the eccentricity of a hyperbola for which distance...
Let be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its the conjugate axis. Then
Options
- A1
- B
- C
- D
Correct answer
B.
Step-by-step solution
Given is the eccentricity of hyperbola and eccentricity of another hyperbola. According to question, \begin{aligned}& 2 \mathrm{a}}_{1} \mathrm{e}}_{1}=2 \times \frac{2 \mathrm{a}_1}{\mathrm{e}_1}\\ \text{ and } \mathrm{e}}_{1}^2=2 \\ \text{ and } \mathrm{e}}_{1}= \pm \sqrt{2} \end{aligned}Here, then Now, \begin{aligned}& \mathrm{a}}_{2}=2 \mathrm{~b}}_{2} \\ \text{ and } \mathrm{~b}}_{2}=\frac{\mathrm{a}_2}{2} \end{aligned}c_2^2=a_2^2+b_2^2\begin{aligned} & \mathrm{c}}_{2}^2=\mathrm{a}}_{2}^2+\frac{\mathrm{a}_2^2}{4}=\frac{5 \mathrm{a}_2^2}{4}\\ \text{ and } \mathrm{c}}_{2}=\frac{\sqrt{5}}{2} \mathrm{a}}_{2} \\ \text{ and } \frac{\mathrm{c}_2}{\mathrm{a}_2}=\frac{\sqrt{5}}{2}\\ \text{ and } \mathrm{e}}_{2}=\frac{\sqrt{5}}{2} \end{aligned}$\mathrm{e}_1 \mathrm{e}_2=\sqrt{2} \times \frac{\sqrt{5}}{2}=\frac{\sqrt{10}}{2}$ So, option (b) is correct.