First slide

Analysis of two circles in circles

Question

The two circles $x^{2} + y^{2} - 2 x - 3 = 0$ and $x^{2} + y^{2} - 4 x - 6 y - 8 = 0$ are such that

Moderate

A

they touch each other
B

they intersect each other
C

one lies inside the other
D

each lies outside the other

Solution

The coordinates of centres $C_{1}$ and $C_{2}$ of two circles are $(1, 0)$ and $(2, 3)$ respective. Let $r_{1}$ and $r_{2}$ be the radii of two circles. Then, $r_{1} = 2$ and $r_{2} = \sqrt{21} .$
Clearly $r_{1} - r_{2} < C_{1} C_{2} < r_{1} + r_{2}$
Hence, the two circles intersect each other.

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