Question

Find the tangent of the angle between the lines whose intercepts on the axes are respectively $a,-b$ and $b,-a$

Answer 1

Given intercepts make lines as

$\frac{x}{a}-\frac{y}{b}=1$ and $\frac{x}{b}-\frac{y}{a}=1$

From above two equations $m_1=\frac{b}{a}$ and $m_2=\frac{a}{b}$...(1)

$\tan \theta =\left|\frac{m_2-m_1}{1+m_{2}m1}\right|$

Using (1) in above equation we get,

$\tan \theta =\left|\frac{\frac{a}{b}-\frac{b}{a}}{1+1}\right|$

$\Rightarrow \tan \theta =\frac{a^2-b^2}{2ab}$