Question

Prove that chord of contact of the pair of tangents to the circle $x^2+y^2=1$ drawn from any point on the line 2x + y = 4 passes through a fixed point. Also, find the coordinates of that point.

Answer 1

Let (a,b) be a point on $2x+y=4$

Then

$2a+b=4$…..(1)

The chord of contact of a pair of tangents from (a,b) is

$ax+by=1$……(2)

Divide (1) by 4 and subtract it from (2)

$a(x-12)+b(y-14)=0$

This shows that $(\frac{1}{2},\frac{1}{4})$always lie on the chord of contact for any value of a and b.