Question

A straight line $L$ is drawn through the point $A(2,1)$ such that its point of intersection with the straight line $x+y=9$ is at distance of $3\sqrt{2}$ from $A$. Find the angle which the line $L$ makes with the positive direction of the x-axis.

Answer 1

Let the point of intersection be $P(h,k)$

As $P$ is on $x+y=9$

$\Rightarrow k=9-h$

$\Rightarrow P(h,9-h)$

Given, $PA=3\sqrt{2}$

$\Rightarrow \sqrt{(h-2{)}^2+(g-h-1{)}^2}=3\sqrt{2}$

$\Rightarrow (h-2{)}^2+(h-8{)}^2=18$

$\Rightarrow 2h^2-20h+50=0$

$\Rightarrow h^2-10h+25=0$

$\Rightarrow (h-5{)}^2=0$

$\Rightarrow h=5\Rightarrow k=4$

$\Rightarrow L$ passes through $(5,4)$ and $(2,1)$

$\Rightarrow$ Slope $=\frac{4-1}{5-2}=\frac{3}{3}=1$

$\Rightarrow \tan \theta =1$

$\Rightarrow \theta =\frac{\pi }{4}$.