Question

Assertion :The chord of contact of tangent 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 $\sqrt{{l_1}^2+{l_2}^2}$ Reason: The equation of chord of contact of tangents from the point $P(x_1,y_1)$ to the circle $x^2+y^2=a^2$ is $x{x}_1+y{y}_1=a^2$

• A. Assertion is true, Reason is true and reason is correct explanation for Statement-1.
• B. Assertion is true, Reason is true and Reason is NOT correct explanation for Assertion.
• C. Assertion is true, Reason is false
• D. Assertion is false, Reason is true

Answer 1

The correct option is A Assertion is true, Reason is true and reason is correct explanation for Statement-1.

Let $P\equiv (x_1,y_1)$ and $Q\equiv (x_2,y_2)$

Let the equation of the given circle be $x^2+y^2=a^2$

The equation of chord of contact of tangents from the point $P(x_1,y_1)$ to the given circle is

$x{x}_1+y{y}_1=a^2$

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

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