Q73 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics
If the distance between the foci and the distance between the two directrixes are in the ratio $3: 2$ for a hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, then a : b is
- A$\sqrt{3}: \sqrt{2}$
- B$2: 1$
- C$1: 2$
- Correct answerD$\sqrt{2}: 1$
Solution
For the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, the distance between the foci is $2ae$ and the distance between the directrices is $\dfrac{2a}{e}$.
Given the ratio of these distances is $3:2$, we have:
$\dfrac{2ae}{2a/e} = \dfrac{3}{2}$
$e^2 = \dfrac{3}{2}$
Using the relation $b^2 = a^2(e^2 - 1)$, we substitute $e^2 = \dfrac{3}{2}$:
$b^2 = a^2 \left( \dfrac{3}{2} - 1 \right) = a^2 \left( \dfrac{1}{2} \right)$
$\dfrac{b^2}{a^2} = \dfrac{1}{2}$
$\dfrac{b}{a} = \dfrac{1}{\sqrt{2}}$
Therefore, $\dfrac{a}{b} = \sqrt{2} : 1$.
Answer: $\sqrt{2}: 1$
Given the ratio of these distances is $3:2$, we have:
$\dfrac{2ae}{2a/e} = \dfrac{3}{2}$
$e^2 = \dfrac{3}{2}$
Using the relation $b^2 = a^2(e^2 - 1)$, we substitute $e^2 = \dfrac{3}{2}$:
$b^2 = a^2 \left( \dfrac{3}{2} - 1 \right) = a^2 \left( \dfrac{1}{2} \right)$
$\dfrac{b^2}{a^2} = \dfrac{1}{2}$
$\dfrac{b}{a} = \dfrac{1}{\sqrt{2}}$
Therefore, $\dfrac{a}{b} = \sqrt{2} : 1$.
Answer: $\sqrt{2}: 1$
Hence, the correct answer is option: 'D'.