If y + 3x = 0 is the equation of a chord of the circle, $x^2+y^2-30x=0$ , then the equation of the circle with this chord as diameter is

$\begin{array}{rcl}\mathrm{Equation} \mathrm{of} \mathrm{circle} \mathrm{is} \mathrm{s}+\mathrm{\lambda }\left(\mathrm{L}\right)& =& 0\\ ({\mathrm{x}}^2+{\mathrm{y}}^2-30\mathrm{x})+\mathrm{\lambda }(3\mathrm{x}+\mathrm{y})& =& 0\to \left(1\right)\\ {\mathrm{x}}^2+{\mathrm{y}}^2-30\mathrm{x}+3\mathrm{\lambda x}+\mathrm{\lambda y}& =& 0\\ {\mathrm{x}}^2+{\mathrm{y}}^2-\mathrm{x}(30-3\mathrm{\lambda })+\mathrm{\lambda y}& =& 0\\ \mathrm{Since} \mathrm{centre} \mathrm{C} & =& \left(\frac{30-3\mathrm{\lambda }}{2},\frac{-\mathrm{\lambda }}{2}\right) \mathrm{is} \mathrm{lies} \mathrm{on} y+3x=0 then \\ & & \\ \frac{-\mathrm{\lambda }}{2}+\frac{3(30-3\mathrm{\lambda })}{2}& =& 0\\ 90-10\mathrm{\lambda }& =& 0\\ & & \boxed{\mathrm{\lambda }=9}\\ from \left(1\right) circle is x^2+y^2-3x+9y& =& 0\\ & & \end{array}$