Question

In an Argand diagram, the point P represents the complex number z, where z = x+iy. Given that z +2 = λi(z +8), where λ is a real parameter, find the Cartesian equation of the locus of P as λ varies. If also z = µ(4 + 3i), where µ is real, prove that there is only one possible position for P.

Answer 1

Given $z +2 = λi(z +8) \implies x+iy+2 = \lambda i(x+iy+8)$

$\implies x+2 + iy = -\lambda y + i\lambda(x+8) \implies x+\lambda y+2=0, \lambda x -y +8\lambda =0$

$\implies (1+\lambda^2)x+(2+8\lambda^2) =0$ and $(\lambda^2 -1) y+10\lambda = 0$

So $x = \frac{2+8\lambda^2}{1+\lambda^2}, y= \frac{10\lambda}{\lambda^2-1}$ is the locus of point P.

If $z=\mu(4+3i)$ then $x= 4\mu \ and \ y = 3\mu \implies \frac{x}{y} = \frac{4}{3}$

Thus there is no parameter in final equation of point, so there is only one possible position for P.