Question

If a line is drawn through a fixed point $P(\alpha ,\beta )$ to cut the circle $x^2+y^2=a^2$ at $A$ and $B$, then find the value of $PA.PB$.

Answer 1

REF. Image

We know that

$PA.PB=(PC{)}^2$ (Tangent seant theorem)

$\therefore O{P}^2=\sqrt{(\alpha -0{)}^2+(\beta -0{)}^2}$

$\therefore O{P}^2={\alpha }^2+{\beta }^2$

$\therefore O{P}^2=P{C}^2+O{C}^2$

$\therefore {\alpha }^2+{\beta }^2=P{C}^2+a^2$

$\therefore P{C}^2={\alpha }^2+{\beta }^2-a^2$