First slide

Definition of a circle

Question

If the tangents are drawn to the circle $x^{2} + y^{2} = 12$ at the point where it meets the circle $x^{2} + y^{2} - 5 x + 3 y - 2 = 0$ then the point of intersection of these tangents, is

Moderate

A

$(6, - 6)$
B

$(6, 18 / 5)$
C

$(6, - 18 / 5)$
D

none of these

Solution

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
$5 x - 3 y - 10 = 0$
Also, the equation of the chord of contact is
$h x + k y - 12 = 0$
Equations (i) and (ii) represent the same line. Therefore,
$\frac{h}{5} = \frac{k}{- 3} = \frac{- 12}{- 10} \Rightarrow h = 6, k = - 18 / 5$
Hence, the required point is $(6, - 18 / 5)$ .

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