The chord of contact of tangents from a point $P$ to a circle passes through $Q$. If$l_1$ and $l_2$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $PQ$is equal to

Let $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ be two points and

$x^2+y^2=a^2$ be the given circle. Then, the chord of contact 0f tangents drawn from $P$ to the given circle is

$x{x}_1+y{y}_1=a^2$ It will pass through $Q\left(x_2, y_2\right)$ if

$x_1{x}_2+y_1{y}_2=a^2$
Now, $l_1=\sqrt{x_1^2+y_1^2-a^2},l_2=\sqrt{x_2^2+y_2^2-a^2}$

$\therefore PQ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

$\Rightarrow PQ=\sqrt{\left(x_2^2+y_2^2\right)+\left(x_1^2+y_1^2\right)-2\left(x_1{x}_2+y_1{y}_2\right)}$

$\begin{array}{l}\Rightarrow PQ=\sqrt{\left(x_2^2+y_2^2\right)+\left(x_1^2+y_1^2\right)-2a^2}[U\mathrm{sing}(\mathrm{i}\left)\right]\\ \Rightarrow PQ=\sqrt{\left(x_1^2+y_1^2-a^2\right)+\left(x_2^2+y_2^2-a^2\right)}\\ \Rightarrow PQ=\sqrt{l_1^2+l_2^2}\end{array}$