Fun with Loci I

The dot product between a vector from $(x,y)$ to $(P_x,P_y)$ and a vector from $(x,y)$ to $(Q_x,Q_y)$ should be zero :

$(Q_x - x)(P_x-x) + (Q_y-y)(P_y-y) = 0 \\ (-7-x)(0-x) + (4-y)(-2-y) = 0 \\ x^2 + y^2 + 7x -2y - 8 = 0$

The locus is a circle with diameter $PQ$ - this is an application of Thales's theorem (or in fact of its converse).

The centre of the circle is the midpoint of $PQ$ (ie $\left(-\frac72,1\right)$ ); the radius is half the distance $PQ$ (ie $\frac{\sqrt{85}}{2}$ ). Its equation is then

$\left(x+\frac72 \right)^2 + (y-1)^2=\frac{85}{4}$

which after simplifying becomes

$\boxed{x^2+y^2+7x-2y-8=0}$