First slide

Analysis of two circles in circles

Question

For the given circles $5 x^{2} + y^{2} - 6 x - 2 y + 1 = 0$ and $x^{2} + y^{2} + 2 x - 8 y + 13 = 0$ which of the following is true?

Moderate

A

One circle lies inside the other
B

One circle lies completely outside the other
C

Two circles intersection in two points
D

They touch each other externally

Solution

The centres of the given circles are $C_{1} (3, 1)$ and $C_{2} (- 1, 4)$ and corresponding radii are $r_{1} = \sqrt{3^{2} + 1^{2} - 1} = 3$
and $r = \sqrt{(- 1)^{2} + 4^{2} - 13} = 2$ respectively.
Now $C_{1} C_{2} = \sqrt{(3 + 1)^{2} + (1 - 4)^{2}} = 5$
Clearly, $C_{1} C_{2} = \sqrt{(3 + 1)^{2} + (1 - 4)^{2}} = 5$
So, given circles touch each other externally.

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