Question

What loci are represented by the equations :
$(y-mx-c{)}^2+(y-m^{\prime }x-c^{\prime }{)}^2=0.$

Answer 1

${(y-mx-c)}^2+{(y-m^{{}^{\prime }}x-c^{{}^{\prime }})}^2=0{(y-mx-c)}^2=-{(y-m^{{}^{\prime }}x-c^{{}^{\prime }})}^2$

As you can see both sides of the equation are perfect squares one of which is positive and the other is negative. This is only possible when both sides are simultaneously zero.

They will be zero simultaneously at the point of intersection of $y-mx-c$ and $y-m^{{}^{\prime }}x-c^{{}^{\prime }}$

Thus the equation represents the point of intesection of both lines .