Question

Find the locus of midpoint of line segment intercepted between real and imaginary axes by the line $a\overline{z}+\overline{a}z+b=0$. where $b$ is a real parameter .

• A. $az+a\overline{z}=0$.
• B. $az=\overline{az}$.
• C. $az+z\overline{a}=0$.
• D. $az+\overline{az}=0$.

Answer 1

The correct option is D $az+\overline{az}=0$.

$\overline{a}z+a\overline{z}+b=0$

Let $z=x+iy$

$\overline{z}=x-iy$

Let $a=c+id$, where $c$ and $d$ are real numbers.

$\overline{a}=c-id$

Put in the equation, we have

$(c-id)(x+iy)+(c+id)(x-iy)+b=0$

$(cx+dy+i(cy-dx\left)\right)+(cx+dy+i(-cy+dx\left)\right)+b=0$

$2cx+2dy+b=0$

Let the mid-point of the line segment to be $(h,k)$

Line intersect the axis at

Real Axis:

$y=0$

$x=-\frac{b}{2d}$

$h=-\frac{b}{4c}$

$k=-\frac{b}{4d}$

$ch=dk$

$h\to x,k\to y$

$y=\frac{c}{d}x$