If the acute angle between lines a x 2 + 2 h x y + b y 2 = 0 is π 4 , then 4 h 2 =

Question

If the acute angle between lines a x 2 + 2 h x y + b y 2 = 0 is &#960; 4 , then 4 h 2 =

Aicharya · Accepted Answer

We know that, the angle between lines of a homogenous pair is

$θ = \tan^{- 1} (\frac{2 \sqrt{h^{2} - a b}}{a + b})$

$\Rightarrow \frac{π}{4} = \tan^{- 1} (\frac{2 \sqrt{h^{2} - a b}}{a + b})$

$\Rightarrow \tan \frac{π}{4} = \frac{2 \sqrt{h^{2} - a b}}{a + b}$

$\Rightarrow \frac{2 \sqrt{h^{2} - a b}}{a + b} = 1$

$\Rightarrow 4 (h^{2} - a b) = {(a + b)}^{2}$

$\Rightarrow 4 h^{2} = a^{2} + b^{2} + 6 a b$

If the acute angle between lines a x 2 + 2 h x y + b y 2 = 0 is &#960; 4 , then 4 h 2 =