Question

The lengths of the tangents from two points $A$ and .$B$ to a circle are $l$ and $l{}^{\prime }$ respectively. If the points are conjugate with respect to the circle, then $(AB{)}^2$ is equal to

• A. $l^2+l{}^{\prime 2}$
• B. $|l^2-l{}^{\prime 2}|$
• C. ${(l+l{}^{\prime })}^2$
• D. $ll{}^{\prime }$

Answer 1

The correct option is A $l^2+l{}^{\prime 2}$

Equation of the circle be $x^2+y^2=a^2.A(x_1,y_1),B(x_2+y_2)$
Since they are conjugate $x_1{x}_2+y_1{y}_2=a^2$
$l^2=x_1^2+y_1^2-a^2,{l^{\prime }}^2=x_2^2+y_2^2-a^2{\left(AB\right)}^2={(x_1-x_2)}^2+{(y_1-y_2)}^2=l^2+{l^{\prime }}^2$