First slide

Analysis of two circles in circles

Question

$The number of common tangents of the circles (x + 3)^{2} + (y - 2)^{2} = 49 and$ $(x - 2)^{2} + (y + 1)^{2} = 4 is$

Moderate

A

0
B

1
C

3
D

4

Solution

$Given circles are (x + 3)^{2} + (y - 2)^{2} = 49 and (x - 2)^{2} + (y + 1)^{2} = 4$
$The given circles have centre at C_{1} (- 2, 2), C_{2} (2, - 1)$
$and redii r_{1} = 7, r_{2} = 2$
$Now C_{1} C_{2} = \sqrt{(2 + 2)^{2} + (- 1 - 2)^{2}}$
$= \sqrt{16 + 9} = 5 = r_{1} - r_{2}$
$∴$ given circles touch internally.
$\Rightarrow$ there is only one common tangent

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