Question

The acute angle between the pair of straight lines passing through $(-6,-8)$ and also through the points which divide the line $2x+y+10=0$ enclosed between coordinate axes in the ratio $1:2:2$ in the direction from the point of intersection with the $x-$axis to the point of intersection with $y-$axis is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{12}$

Answer 1

The correct option is C $\frac{\pi }{4}$


$2x+y+10=0$
Intercept form $\frac{x}{-5}+\frac{y}{-10}=1$
coordinates of intercepts

$(-5,0)(0,-10)$

Slope of $2x+y+10=0$ is $-2$

Length=$\sqrt{25+100}=5\sqrt{5}$

Coordinates of $P(-4,-2)Q(-2,-6)$

Slope of$AP=\frac{-2+8}{-4+6}=3$

Slope of$AQ=\frac{-6+8}{-2+6}=\frac{1}{2}$

$\Rightarrow \tan \theta =\frac{3-\frac{1}{2}}{1+\frac{3}{2}}=1\Rightarrow \frac{\pi }{4}=\theta$