Q.If $y=2x$ is a chord of the circle $x^2+y^2-10x=0$, find the equation of the circle with this chord as diameter.

Given the two endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ of a diameter of a circle, the equation of the circle can be written down directly, without first locating the centre or computing the radius, using the fact that an angle inscribed in a semicircle is always a right angle: for any other point $P(x,y)$ on the circle, $\angle APB=90^\circ$, so the slopes of $AP$ and $BP$ multiply to $-1$. Turning that perpendicularity condition into an equation and simplifying gives the diameter form

$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$ …