Question

The chord of contacts of tangents from a point P to a circle passes through Q,if $l_1$ and $l_2$ are the tangents passing from P and Q to the circle ,then PQ is equal to

• A. $\frac{l1+l2}{2}$
• B. $\frac{l1-l2}{2}$
• C. $\sqrt{l_1^2+l_2^2}$
• D. $\sqrt{l_1^2-l_2^2}$

Answer 1

The correct option is C $\sqrt{l_1^2+l_2^2}$

Let $P\equiv (x_1,y_1)$ and $Q\equiv (x_2,y_2)$
Let the equation of given circle be $x^2+y^2=a^2$
The equation of chord of contact of tangent drawn from the point $P(x_1,y_1)$ to the given circle is ${xx}_1+{yy}_1=a^2$
Since it passes through $Q(x_2,y_2)$
$\therefore x_1{x}_2+y_1{y}_2=a^2$ ...(1)
Now $l_1=\sqrt{x_1^2+y_1^2-a^2},l_2=\sqrt{x_2^2+y_2^2-a^2}$
and
$PQ=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}=\sqrt{(x_1^2+y_1^2)+(x_2^2+y_2^2)-2(x_1{x}_2+y_1{y}_2)}=\sqrt{(x_1^2+y_1^2)+(x_2^2+y_2^2-a^2)}=\sqrt{(x_1^2+y_1^2-a^2)(x_2^2+y_2^2-a^2)}=\sqrt{l_1^2+l_2^2}$