Name: Basic Geometrical Concepts of Complex Number Plane
Uploaded: 2019-07-19
Duration: 5 min 49 s
Description: Before studying the geometrical applications of complex numbers, it is essential to recall the basic geometrical concepts of the complex number plane. Let's...

Question

On an Argand diagram, sketch the locus

| z | = | z + 8 i | .

Find the Cartesian equation of this locus.

Accepted Answer

Given $| z | = | z + 8 i |$

$|z - z_{1}| = |z - z_{2}|, |z - z_{1}| < |z - z_{2}|, |z - z_{1}| \leq |z - z_{2}|$ where $z_{1}$ is the fixed point $a + b i$ and $z_{2}$ is the fixed point $c + d i$ and $P (x, y)$ is a moving point on the Argand diagram.

when $|z - z_{1}| = |z - z_{2}|$ , the point $P (x, y)$ moves so that it is always the same distance from $Q (a, b)$ as it is from $R (c, d)$ . The locus is, therefore, the perpendicular bisector of the line $Q R$ .

$| z | = | z + 8 i |$

By comparing $| z | = | z + 8 i |$ with $|z - z_{1}| = |z - z_{2}|$ , we get

$z_{1} = 0 + i 0$ and $z_{2} = 0 - 8 i$

Therefore, $z_{1}$ has Coordinates $(0, 0)$ and $z_{2}$ has Coordinates $(0, - 8)$

The locus is the perpendicular bisector of the points $(0, 0), (0, - 8)$ .

The Midpoint of the line joining points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .

Midpoint of $Q (0, 0)$ and $R (0, - 8)$ is $m = (\frac{0 + 0}{2}, \frac{0 - 8}{2})$

$\Rightarrow m = (0, - 4)$

Slope of a line passing through points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

Slope of a $Q R$ is $\infty$ and so perpendicular bisector has gradient $0$ .

Therefore, perpendicular bisector has equation $y + 4 = 0 (x - 0)$

$\Rightarrow y = - 4$

Cartesian equation of locus is $y = - 4$

The Argand diagram is as follows

On an Argand diagram, sketch the locus $| z | = | z + 8 i | .$ Find the Cartesian equation of this locus.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

On an Argand diagram, sketch the locus |z|=|z+8i|. Find the Cartesian equation of this locus.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

On an Argand diagram, sketch the locus $| z | = | z + 8 i | .$ Find the Cartesian equation of this locus.