Question

The acute angle between two straight lines passing through the point $M(-6,-8)$ and the points in which the line segment $2x+y+10=0$ enclosed between the co-ordinate axes is divided in the ratio $1:2:2$ in the direction from the point of its intersection with the $x$-axis to the point of intersection with the $y$-axis is :

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

Answer 1

Answer: (B)

$\frac{\pi }{4}$

$AP:PQ:QB=1:2:2$

$A(-5,0)---P---Q---B(0,-10)$

$m\equiv (-6,-8)$

$P\equiv (\frac{-20}{5},\frac{-10}{5})\Rightarrow P\equiv (-4,-2)$

$Q\equiv (-2,-6)$

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

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

Angle between PM and QM$\Rightarrow \left|\frac{3-\frac{1}{2}}{1+\frac{3}{2}}\right|=\tan \theta$

$\Rightarrow \theta ={45}^{\circ }=\frac{\pi }{4}$