If the two circles, $x^2 + y^2 + 2 g_1x + 2 f_1y = 0\, \& \,x^2 + y^2 + 2 g_2x + 2 f_2y = 0$ touch each then:
$f_1 g_1 = f_2 g_2$
$\frac{{{f_1}}}{{{g_1}}} = \frac{{{f_2}}}{{{g_2}}}$
$f_1 f_2 = g_1 g_2$
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The equation of the circle which touches the circle ${x^2} + {y^2} - 6x + 6y + 17 = 0$ externally and to which the lines ${x^2} - 3xy - 3x + 9y = 0$ are normals, is
Two circle ${x^2} + {y^2} = ax$ and ${x^2} + {y^2} = {c^2}$ touch each other if
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If one common tangent of the two circles $x^2 + y^2 = 4$ and ${x^2} + {\left( {y - 3} \right)^2} = \lambda ,\lambda > 0$ passes through the point $\left( {\sqrt 3 ,1} \right)$, then possible value of $\lambda$ is