A circle C touches the line x = 2y at the point (2, 1) and...

Family of circle touching line $2y = x$ at point \left(2, 1\right)$<br />$( x - 2{)}^{2} + ( y - 1{)}^{2} + \lambda ( x - 2y ) = 0 . . . ( 1 )$<br />common chord$PQ$is<br />$( x - 2{)}^{2} + ( y - 1{)}^{2} + \lambda ( x - 2y ) - {x}^{2} - {y}^{2} - 2y + 5 = 0$<br />(diameter of${C}_{1}$passes through$( 0, - 1 )$)<br />$4 + 4 + 2\lambda - 1 + 2 + 5 = 0$</p><p>$\lambda = - 7$put in equation \left(1\right)$

$( x - 2{)}^{2} + ( y - 1{)}^{2} - 7 ( x - 2y ) = 0$

${x}^{2} + {y}^{2} - 11x + 12y + 5 = 0$

$r = \sqrt{\dfrac{121}{4} + 36 - 5} = \sqrt{\dfrac{121 + 124}{4}} = \sqrt{\dfrac{245}{4}} = \dfrac{\sqrt{245}}{2} ⇒ \text{diameter} = \sqrt{245}= 7\sqrt{5}$