Question

If the distance between the foci and the distance between the two directrices of the hyperbola x^2/a^2 - y^2/b^2 = 1 are in the ratio 3 : 2, then

b.a is :

Answer 1

please check
i guess answe is 1/√2

Answer 2

a : b = 1 : √2

Step-by-step explanation:

The eccentricity of a hyperbola with the equation $\frac{x^{2} }{a^{2}} - \frac{y^{2} }{b^{2}} = 1$ is given by $e = \sqrt{1 + \frac{b^{2} }{a^{2} } }$

Now, the distance between the foci is 2ae = $2 \sqrt{a^{2} + b^{2}}$

And the distance between the two directrices = $2\times \frac{a}{e} = \frac{2a^{2} }{\sqrt{a^{2} + b^{2} } }$

If the ratio between the distance between two foci and the distance between two directrices is 3 : 2

Then, $\frac{\sqrt{a^{2} + b^{2}}}{\frac{a^{2} }{\sqrt{a^{2} + b^{2}}}} = \frac{3}{2}$

⇒ $\frac{a^{2} + b^{2}}{a^{2}} = \frac{3}{2}$

⇒ a² = 2b²

⇒ a : b = 1 : √2 (Answer)