Question

A straight line $L_1:x-2y+10=0$ meets the circle with equation $x^2+y^2=100$ at $B$ in the first quadrant. If another line $L_2$ through $B$ is perpendicular to $L_1$ and cuts $y-$axis at $P(0,t),$ then the value of $t$ is

Answer 1

Line $L_1:x-2y+10=0$
Perpendicular line to $L_1$ is $2x+y+\lambda =0$
Now, as $\angle B$ is ${90}^{\circ },$ so $L_2$ must pass through $(10,0)$
$20+0+\lambda =0\Rightarrow \lambda =-20$
Now, $2x+y-20=0$
$\Rightarrow 0+t-20=0\therefore t=20$