Two circles each of radius 5 units touch each other at the point (...

Let a point circle: ${( x - 1 )}^{2} + {( y - 2 )}^{2} = 0$

Equation of circle passing through the intersection of the line and the circle:${C}_{1} + \lambda{L}_{1} = 0$

$( x - 1{)}^{2} + ( y - 2{)}^{2} + \lambda ( 4x + 3y - 10 ) = 0$
${x}^{2} + {y}^{2} + ( 2\lambda - 1 ) 2x + \left(\dfrac{3}{2}\lambda - 2\right)2y + 5 - 10\lambda = 0$

$r = \sqrt{{g}^{2} + {f}^{2} - c}$
$r = \sqrt{( 2\lambda - 1{)}^{2} + {\left(\dfrac{3}{2}\lambda - 2\right)}^{2} - ( 5 - 10\lambda )} = 5$
$4{\lambda}^{2} + 1 - 4\lambda + \dfrac{9}{4}{\lambda}^{2} + 4 - 6\lambda - 5 + 10\lambda = 25$
$\dfrac{25}{4}{\lambda}^{2} - 25 = 0$
$\lambda = \pm 2$
for $\lambda = 2$
${x}^{2} + {y}^{2} + 6x + 2y - 15 = 0; {C}_{1} ( - 3, - 1 )$
for $\lambda = - 2$
${x}^{2} + {y}^{2} - 10x - 10y + 25 = 0; {C}_{2} ( 5, 5 )$
$| ( \alpha + \beta ) ( \gamma + \delta ) | = 40$