First slide

Analysis of two circles in circles

Question

The point from which the tangents to the circles
$\begin{array}{l} x^{2} + y^{2} - 8 x + 40 = 0 \\ 5 x^{2} + 5 y^{2} - 25 x + 80 = 0 \\ x^{2} + y^{2} - 8 x + 16 y + 160 = 0 \end{array}$
are equal in length, is

Moderate

A

(8, 15/2)
B

(- 8, 15/2)
C

(8, -15/2)
D

none of these

Solution

The required point is the radical centre of the three given circles. The radical axes of these three circles taken in pairs are:
$3 x - 24 = 0, 16 y + 120 = 0$ and $- 3 x + 16 y + 80 = 0$
Solving any two of these three equations, we get
$x = 8, y = - \frac{15}{2}$
Hence, the required point is (8, -15/2)

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