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 representing complex numbers $z$ satisfying $|z + i| = 1$ and the locus representing complex numbers $w$ satisfying $\arg (w - 2) = \frac{3}{4} π$ . Find the least value of $|z - w|$ for points on these loci.

Accepted Answer

Given that complex numbers $z$ satisfying $|z + i| = 1$ and the locus representing complex numbers $w$ satisfying $\arg (w - 2) = \frac{3}{4} π$ .

Let the complex number $z = x + i y$ , where $x$ and $y$ are real, the locus of the point $M (x, y)$ that satisfies the $|z - z_{1}| = r, |z - z_{1}| < r, |z - z_{1}| \leq r$ where $r$ is a constant and $z_{1}$ is the fixed point $a + b i$ where $a$ and $b$ are real.

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

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

Therefore, the centre is $(0, - 1)$

$∴ | z + i | = 1$ is a circle with centre $(0, - 1)$ and radius $1$ .

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

Given $\arg (w - 2) = \frac{3 π}{4}$

If $\arg (z - z_{1}) = α$ , 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 $α$ .

$\arg (z - z_{1})$ 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 $α$ .

$\Rightarrow \arg (w - 2) = \frac{3 π}{4}$

Comparing with $\arg (z - z_{1}) = α$ , we get

$\Rightarrow z_{1} = 2$

Therefore, the fixed point is $(2, 0)$

Therefore, the locus is half-line from $(2, 0)$ at an angle of $\frac{3 π}{4}$ radians.

From above information we can draw Argand diagram

$|z - w|$ is the distance between the points representing the complex numbers $z$ and $w$ .

The dotted line shows a direction that is perpendicular to the line by passes through the circle.

The minimum distance between the half line and circle is the distance between the centre of the circle and the point where the dotted line meets the half line, less the radius of the circle.

note that $∆ A B C, ∆ A O D and ∆ D E F$ with $45 °, 45 °, 90 °$ and with $| A C | = | A O | = | O D | = 1, | E D | = | E O | - | O D | = 2 - 1 = 1$

The least value of $|z - w| = |C F|$

$= |C D| + |D F|$

$= |A D| - |A C| + \sin \frac{π}{4} |D E|$

$= 2 \times \frac{1}{\sqrt{2}} - 1 + \frac{1}{\sqrt{2}}$

$= \frac{3 \sqrt{2}}{2} - 1$

On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z + i| = 1$ and the locus representing complex numbers $w$ satisfying $\arg (w - 2) = \frac{3}{4} π$ . Find the least value of $|z - w|$ for points on these loci.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

On an Argand diagram, sketch the locus representing complex numbers z satisfying z+i=1 and the locus representing complex numbers w satisfying arg(w-2)=34π. Find the least value of z-w for points on these loci.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z + i| = 1$ and the locus representing complex numbers $w$ satisfying $\arg (w - 2) = \frac{3}{4} π$ . Find the least value of $|z - w|$ for points on these loci.