First slide

Analysis of two circles in circles

Question

There are two circles whose equations are x2+y2=9 and $x^{2} + y^{2} - 8 x - 6 y + n^{2} = 0, n \in I$ . If the two circles having exactly two common tangents, then the number of the possible values of ‘n’ is

Moderate

A

2
B

7
C

8
D

9

Solution

$C_{1} (0, 0)$                            $C_{2} (4, 3)$
$r_{1} = 3 r_{2} = \sqrt{25 - n^{2}}$
-5<n<5    $25 - n^{2} > 0$
For exactly two common tangents
$r_{1} + r_{2} > c_{1} c_{2} \Rightarrow 3 + \sqrt{25 - n^{2}} > 5$
$\Rightarrow n^{2} < 21$
$\Rightarrow - \sqrt{21} < n < \sqrt{21} \dots \dots (2)$
$From (1) and (2), we get - \sqrt{21} < n < \sqrt{21}$
$But n \in I, So n = - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4$

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