Show that the circles.
$x^{2} + y^{2} + 2 a x + 4 a y - 3 a^{2} = 0$
and, $x^{2} + y^{2} - 8 a x - 6 a y + 7 a^{2} = 0$
touch then determine point of contact.

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Solution

$C_{1}$ ( - a , -2a), $r_{1} = 2 \sqrt{2 a}, C_{2} (4 a, 3 a), r_{2} = 3 \sqrt{2 a}$
clearly $C_{1} C_{2} = 5 \sqrt{2 a} = r_{1} + r_{2}$ .
Hence they touch.
The point of contact will divide $C_{1} C_{2}$ in the ratio of radii i . e. 2 : 3. Hence by ratio formula it is (a , 0)

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