First slide

Analysis of two circles in circles

Question

The number of common tangents to the circles
$x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is

Moderate

A

3
B

4
C

1
D

2

Solution

The coordinates of the centres and radii of the given circles are:
$C_{1} : (2, 3)$ $C_{2} : (- 3, - 9)$
$r_{1} : 5$ $r_{2} : 8$
we find that $C_{1} C_{2} = 13 = r_{1} + r_{2}$ So, the two circles touch each other externally. Hence, there are 3 common tangents.

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