Question

If the equation of the common tangent at the point $(1,-1)$ to the two circles, each of radius $13$, is $12x+5y-7=0$, then the centers of the two circles are

• A. $(13,4)$
• B. $13,-4)$
• C. $(-11,6)$
• D. $(-11,-6)$

Answer 1

Answer: (A), (D)

The correct options are
A $(13,4)$
D $(-11,-6)$

Let $A,B$ be the centers of the two circles.

Slope of the common tangent $=-\frac{12}{5}$

$\therefore$ Slope of $AB$ is $\tan \theta =-\frac{1}{-\frac{12}{5}}=\frac{5}{12}$

The point $(1,-1)$ lies on the line $AB$ and the points $A$ and $B$ are at a distance $13$ from the point $(1,-1)$.

$\therefore$ Coordinates of $A$ and $B$ are $(1\pm 13\cos \theta ,-1\pm 13\sin \theta )$,

where $\tan \theta =\frac{5}{12}$

$\Rightarrow (1\pm 13.\frac{12}{13},-1\pm 13\frac{5}{13})\Rightarrow (1\pm 12,-1\pm 5)$

$\Rightarrow (13,4)$ or $(-11,-6)$