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

Sketch the loci

\arg (z + 4 + 2 i) = \frac{5 π}{6}

and

| z - 5 i | = 4

on an Argand diagram. Determine whether or not there is a complex number,

z

, that satisfies both loci.

Accepted Answer

Given $| z - 5 i | = 4$

If $|z - z_{1}| = r, |z - z_{1}| < r, |z - z_{1}| \leq r$ where $r$ is a constant and $z_{1}$ is the fixed point $a + b i$ where $a$ and $b$ are real.

$|z - z_{1}| = r$ is the circumference of a circle with centre $(a, b)$ and radius $r$ .

Now comparing the given $| z - 5 i | = 4$ with $|z - z_{1}| = r$ we get $z_{1} = 0 + 5 i$

Therefore, the centre is $(0, 5)$

$∴ | z - 5 i | = 4$ is a circle with centre $(0, 5)$ and radius $4$ .

It is only the circle, not the points within the circle.

The circle is drawn as a solid line as it is included.

Given $\arg (z + 4 + 2 i) = \frac{5 π}{6}$

If $a r g (z - z_{1}) = α$ , where $α$ is a fixed angle and $z_{1}$ is the fixed point $(a + b i)$ , then this represents a half line from $(a, b)$ with an angle $α$ .

$a r g (z - z_{1})$ is the angle made between the direction parallel to the real axis and the direction of the vector which has point $Q (a, b)$ and an unknown point $P (x, y)$ .

Points below $Q (a, b)$ are not part of the locus as the angle made between these points and the relevant horizontal line is $π - α$ , not $α$ .

$\Rightarrow \arg (z + 4 + 2 i) = \frac{5 π}{6}$

Comparing with $a r g (z - z_{1}) = α$ we get

$z - z_{1} = z + 4 + 2 i$

$\Rightarrow z - z_{1} = z - (- 4 - 2 i)$

$\Rightarrow z_{1} = - 4 - 2 i$

Therefore, the fixed point is $(- 4, - 2)$

Therefore, the locus is half-line from $(- 4, - 2)$ at an angle of $\frac{5 π}{6}$ radians.

From above information we can draw below an Argand diagram

From above diagram we can say that there is no complex number that satisfies both loci because there is no intersection of between the locus.

Sketch the loci $\arg (z + 4 + 2 i) = \frac{5 π}{6}$ and $| z - 5 i | = 4$ on an Argand diagram. Determine whether or not there is a complex number, $z$ , that satisfies both loci.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Sketch the loci arg(z+4+2i)=5π6 and |z-5 i|=4 on an Argand diagram. Determine whether or not there is a complex number, z, that satisfies both loci.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Sketch the loci $\arg (z + 4 + 2 i) = \frac{5 π}{6}$ and $| z - 5 i | = 4$ on an Argand diagram. Determine whether or not there is a complex number, $z$ , that satisfies both loci.