Question

The angle between the lines represented by the equation $2x^2+3xy-5y^2=0$ is :

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. ${\tan }^{-1}\left|\begin{array}{c}\frac{12}{5}\end{array}\right|$
• D. ${\tan }^{-1}\left|\begin{array}{c}\frac{7}{3}\end{array}\right|$

Answer 1

The correct option is D ${\tan }^{-1}\left|\begin{array}{c}\frac{7}{3}\end{array}\right|$

General form of an equation representing a pair of lines is $a{x}^2+2hxy+b{y}^2=0$ .... $\left(i\right)$

Given equation is $2x^2+3xy-5y^2=0$ .... $\left(ii\right)$

Comparing $\left(ii\right)$ with $\left(i\right),$ we get

$a=2,h=\frac{3}{2},b=-5$
Angle between the lines is given by

$\tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$
$\therefore \tan \theta =\left|\frac{2\sqrt{{\left(\frac{3}{2}\right)}^2+10}}{2-5}\right|=\left|\frac{\sqrt{49}}{-3}\right|=\left|\frac{7}{3}\right|$
$\Rightarrow \theta ={\tan }^{-1}\left|\frac{7}{3}\right|$