Question

If the lengths of the tangents from two points A, B to a circle are 6, 7 respectively. If A, B are conjugate points then AB =

• A. 5
• B. $\sqrt{85}$
• C. $\frac{\sqrt{85}}{2}$
• D. none

Answer 1

The correct option is B $\sqrt{85}$

Let the equation of the circle be $x^2$ + $y^2=a^2$ and A $(x_1,y_1)$, $B(x_2,y_2)$
A, B are conjugate points $\Rightarrow x_1{x}_2+y_1{y}_2=a^2$ $\to$ (1)
Length of the tangents from A, B are 6, 7 respectively $\Rightarrow$ $\sqrt{x_1^2+y_1^2-a^2}$ = 6, $\sqrt{x_2^2+y_2^2-a^2}$= 7
$\Rightarrow$ $x_1^2+y_1^2-a^2$ = 36
$x_2^2+y_2^2-a^2$ = 49
$\Rightarrow$ $x_1^2+y_1^2+x_1^2+y_2^2-{2a}^2$ = 85
$\Rightarrow$ $x_1^2+y_1^2+x_1^2+y_2^2-2$
(x${}_1$ x${}_2$ + y${}_1$ y${}_2$) = 85 [From (1)]
$\Rightarrow$ $(x_1^2+x_2^2-2x_1{x}_2)$ + $(y_1^2+y_2^2-2y_1{y}_2)$ = 85 $\Rightarrow$ $(x_1-x_2{)}^2$ + $(y_1-y_2{)}^2$ = 85 $\Rightarrow$ $A{B}^2$ = 85 $\Rightarrow$ AB = $\sqrt{85}$