Question

If tangents are drawn to the circle $x^2+y^2=12$ at the points where it intersects the circle $x^2+y^2-5x+3y-2=0$, then the coordinates of the point of intersection of those tangents are

• A. $\left(-6,\frac{18}{5}\right)$
• B. $\left(6,\frac{18}{5}\right)$
• C. $\left(-6,\frac{-18}{5}\right)$
• D. $\left(6,\frac{-18}{5}\right)$

Answer 1

Answer: (D)

Let $(h,k)$ be the point of intersection of the
tangents. Then, the chord of contact of tangents is the common chord of the circles $x^2+y^2=12$ and $x^2+y^2-5x+3y-2=0$

The equation of the common chord is
$5x-3y-10=0\dots$(i)
Also, the equation of the chord of contact is
$hx+ky-12=0\dots$ (ii)
Eqs. (i) and (ii) represents the same line.
Therefore,
$\frac{h}{5}=\frac{k}{-3}=\frac{-12}{-10}$
$\Rightarrow h=6,k=\frac{-18}{5}$
Hence, the required point is $\left(6,-\frac{18}{5}\right)$.