Question

Find the condition for the line $px+qy+r=0$ to touch the circle $x^2+y^2=a^2$ . Also find the co-ordinates of the point of contact.

Answer 1

Let the line px + qy + r = 0 ..(1)
touch the circle x${}^2$ + y${}^2$ = a${}^2$ .(2)
at the point (h, k).
Now the equation of tangent at (h, k) is
xh + yk - a${}^2$ = 0 .(3)
Comparing (1) and (3), we get $\frac{h}{p}=\frac{k}{q}=\frac{-a^2}{r}$
whence h = -a${}^2$ p/r and k = a${}^2$ q/r
Since (h , k) lies on the given circle $\therefore$ from (2)
a${}^4$ p${}^2$/r${}^2$ + a${}^4$ q${}^2$/r${}^2$ a${}^2$
or r${}^2$ = a${}^2$ (p${}^2$ + q${}^2$)
which is the required condition and the point of
contact is (- a${}^2$ p/r , - a${}^2$ q/r).