Question

Consider $\Delta ABC$ in Argand plane. Let $A\left(0\right),B\left(1\right)$ and $C(1+i)$ be its vertices and $M$ be the mid-point of $CA$. Let $z$ be a variable complex number on the line $BM$. Let $u$ be another variable complex number defined as $u=z^2+1$.
Locus of $u$ is

• A. parabola
• B. ellipse
• C. hyperbola
• D. none of these

Answer 1

Answer: (A)

parabola

$BM=y-0=-1(x-1)$

$x+y=1$

$\therefore$ $\sqrt{u-1}=1+i(1-t)$

$u=2t+2it(1-t)$

$x=2t$ and $y=2t(1-t)$

$y=x(1-\frac{x}{2})$

$2y=2x-x^2$

$\Rightarrow$ $(x-1{)}^2=-2(y-\frac{1}{2})$

which is a parabola. Its axis is $x=1$, i.e., $z+\overline{z}=2$ and directrix is $y=1$, i.e., $z-\overline{z}=2i$.