Question

The chord of the contact of tangents from a point $P$ to a circle passes through $Q$ ($Q$ lies outside the circle). If $a_1$ and $a_2$ are the lengths of the tangents from $P$ and $Q$ to the circle respectively, then $PQ$ is equal to

• A. $\sqrt{a_1^2+a_2^2}$
• B. $a_1^2+a_2^2$
• C. $a_1+a_2$
• D. $\frac{a_1+a_2}{2}$

Answer 1

The correct option is A $\sqrt{a_1^2+a_2^2}$

Let $P\equiv (x_1,y_1)$ and $Q\equiv (x_2,y_2)$
and equation of the circle is $x^2+y^2=a^2$
Now the eqaution of the chord of the contact of tangents drawn from the $P$ is
$x{x}_1+y{y}_1=a^2$
this line also passes through $Q$
$\therefore x_1{x}_2+y_1{y}_2=a^2$
Now $a_1=\sqrt{x_1^2+y_1^2-a^2}$
$a_2=\sqrt{x_2^2+y_2^2-a^2}$
$PQ=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}=\sqrt{x_1^2+y_1^2+x_2^2+y_2^2-2(x_1{x}_2+y_1{y}_2)}=\sqrt{a_1^2+a_2^2+2a^2-2a^2}=\sqrt{a_1^2+a_2^2}$