If the distance between the focii and the distance between the directrices of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) are in the ratio \(3: 2\), then \(a: b\) is

Given equation of hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)
Now, distance between the focii \(=2 a e\) and distance between the directrices \(=\frac{2 a}{e}\) (by condition)
Given, ratio \(=3: 2\)
\(\frac{2 a e}{(2 a / e)}=\frac{3}{2}\)
\(\Rightarrow \quad 4 a e^{2}=6 a\)
\(\Rightarrow \quad e^{2}=\frac{3}{2} \Rightarrow e=\sqrt{\frac{3}{2}}...(i)\)
\(\because\) Eccentricity of hyperbola \(=\sqrt{\frac{a^{2}+b^{2}}{a^{2}}}\)
\(\therefore \quad \sqrt{\frac{a^{2}+b^{2}}{a^{2}}}=\sqrt{\frac{3}{2}} \quad\) [from Eq. (i)]
\(\Rightarrow \quad \frac{a^{2}+b^{2}}{a^{2}}=\frac{3}{2}\) (squaring on both sides)
\(\Rightarrow \quad 1+\frac{b^{2}}{a^{2}}=\frac{3}{2}\)
\(\Rightarrow \quad\left(\frac{b}{a}\right)^{2}=\frac{1}{2}\)
\(\therefore \quad \frac{b}{a}=\frac{1}{\sqrt{2}}\)
Hence, \(\quad a: b=\sqrt{2}: 1\)