Question

The equation $x^2-5xy+p{y}^2+3x-8y+2=0$ represents a pair of straight lines. If $\theta$ is the acute angle between them, then $\sin \theta$ equals

• A. $\frac{1}{\sqrt{50}}$
• B. $\frac{1}{7}$
• C. $\frac{1}{5}$
• D. $\frac{1}{\sqrt{10}}$

Answer 1

The correct option is A $\frac{1}{\sqrt{50}}$

For pair of straight lines,
$\Delta =0$
$\Rightarrow \left(1\right)\left(p\right)\left(2\right)+2(-4)\left(\frac{3}{2}\right)\left(\frac{-5}{2}\right)-1(-4{)}^2-p{\left(\frac{3}{2}\right)}^2-2{\left(\frac{-5}{2}\right)}^2=0$
$\Rightarrow 2p+30-16-\left(\frac{9}{4}\right)p-\left(\frac{25}{2}\right)=0$
$\Rightarrow p=6$

$\tan \theta =\frac{2\sqrt{h^2-ab}}{|a+b|}=\frac{2\sqrt{\frac{25}{4}-6}}{|1+6|}=\frac{1}{7}\therefore \sin \theta =\frac{1}{\sqrt{50}}$