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 $x^{2}+y^{2}-5 x+3 y-2=0$

The equation of the common chord is
$5 x-3 y-10=0 \ldots $(i)
Also, the equation of the chord of contact is
$ h x+k y-12=0 \quad \ldots $ (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 $\left(6,-\frac{18}{5}\right)$.