Question

Find the measure of acute angle between the lines represented by
$2x^2+7xy+3y^2=0$

Answer 1

Comparing the equation
$2x^2+7xy+3y^2=0$ with
$a{x}^2+2hxy+b{y}^2=0,$ we get,
$a=2,2h=7,$ i.e.$h=\frac{7}{2}$ and $b=3.$
Let $\theta$ be the acute angle between the lines.
$\therefore \tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$
$=\left|\frac{2\sqrt{{\left(\frac{7}{2}\right)}^2-2\left(3\right)}}{2+3}\right|$
$=\left|\frac{2\sqrt{\left(\frac{49}{4}\right)-6}}{5}\right|$
$=\left|\frac{2\sqrt{\left(\frac{49-24}{4}\right)}}{5}\right|$
$=\left|\frac{2\sqrt{\left(\frac{25}{4}\right)}}{5}\right|$
$=\frac{2\times \left(\frac{5}{2}\right)}{5}$
$=\frac{5}{5}$
$\tan \theta =1$
$\therefore \theta =\tan 1={45}^o$
$\therefore \theta ={45}^o$