If y + 3x = 0 is the equation of a chord of the circle x ^ 2 + y ^...

To find point of intersection of given chord $y = - 3x$ with the given circle ${x}^{2} + {y}^{2} - 30x = 0$ eliminate $y$ from two equations
Point of intersection
$⇒$ ${x}^{2} + 9{x}^{2} - 30x = 0$
$⇒$ $10x\left(x - 3\right) = 0$
$⇒$ $x = 0$ or $x = 3$
Therefore, $y = 0$ and $y = - 9$ are the corresponding ordinates.
Point of intersection $( 0, 0 )$, $( 3, - 9 )$
Diametric form of equation of circle,
$x\left(x - 3\right) + y\left(y + 9\right) = 0$
$⇒ {x}^{2} + {y}^{2} - 3x + 9y = 0$ .