Indian Statistical Institute B.Math & B.Stat : Complex Numbers

Show that the set of complex numbers $z$ satisfying the equation $(3+7i)z+(10-2i)\overline{z}+100 = 0$ represents, in the Argand plane, a point. Let $z=x+iy$, taking the conjugate of the given equation we have $(3-7i)\overline{z}+(10+2i)z+100 = 0$ Adding the two equations we get, $26x-18y+200 = 0$ (do the calculations yourself!), this shows that $z$ lies on the line $26x-18y+200 = 0$ Subtracting the two equations we get, $10x-4y = 0$, this again shows that that $z$ lies on the line $10x-4y = 0$ Thus $z$ satisfies both the equations $26x-18y+200 = 0$ and $10x-4y = 0$, thus $z$ represents a point in the Argand Plane.