In 
, how could you write the distance between the two points??
$$ |z_2 - z_1| $$
This result is a case of what result for vectors??
$$ \vec{AB} = b - a $$
For two complex numbers $z_1$ and $z_2$, how could you write the distance between them??
$$ |z_2 - z_1| $$
If $z$ is a variable representing any complex number and $z_1$ is a fixed point, what is $|z - z_2|??
The distance from $z$ to $z_2$.
What is |z_2 - z_1| for two complex numbers??
The distance between $z_2$ and $z_1$.
The equation $|z - z_1| = r$ means what in practical terms??
The distance between $z_1$ and $z$ is fixed at $r$. This describes a circle with radius $r$.
If $z_1 = a+bi$ and $|z-z_1| = r$, what is the radius of the circle formed??
$$ r $$
If $z_1 = a+bi$ and $|z-z_1| = r$, where is the centre of the circle??
$$ (a, b) $$
How could you rewrite $|z - 4 + 8i|$ more usefully?
$$ |z - (4 - 8i)| $$
Where is the centre of $|z - (1 + 2i)| = 4$??
$$ (1, 2) $$
What is the radius of $|z - (1 + 2i)| = 4$??
$$ 4 $$
Where is the centre of $|z - 1 + 2i| = 4$??
$$ (1, -2) $$
If $z = x + yi$ and $z_1 = x_1 + y_1 i$, how could you rewrite $|z - z_1|$??
$$ | (x - x-1) + i(y - y1) | $$
What do you get if you square both sides of $| (x - x-1) + (y - y1) | = r$??
$$ (x - x-1)^2 + (y - y1)^2 = r^2 $$
What is $(x - x-1)^2 + (y - y1)^2 = r^2$??
The Cartesian equation of a circle.
describes what relationship??
$$ |z - z_1| = r $$
describes what relationship??
$$ |z - z_1| = |z - z_2| $$
What is the locus of points that are an equal distance from two different points $z_1$ and $z_2$??
The perpindicular bisector of the line segment goining the two points.
What is $|z - z_1| = |z - z_2|$??
The locus of points forming a perpindicular bisector of the line segment joining $z_1$ and $z_2$.
What is the equation of the following circle 
??
$$ |z - (5 + 4i)| = 8 $$
What are the points in where $\text{Re}(z) = 0$??
What lines could you visualise in in order to find the points of intersection with the imaginary axis??
What are the lengths of the line segments??
What is the length of the unlabeled side length??
$$ \sqrt{8^2 - 5^2} $$
How could you find the length of the unlabelled side??
Using Pythagoras.
If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the top point??
$$ 4 + \sqrt{8^2 - 5^2} $$
If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the bottom point??
$$ 4 - \sqrt{8^2 - 5^2} $$
What does the Cartesian equation for a perpindicular bisector look like??
$$ y = mx + c $$
How can you find the Cartesian equation for a perpindicular bisector??
- Find the equation of the lines passing through two points
- Find the perpindicular line at the midpoint
This the relationship $|z - (a + bi)| = r|$. What line could you visualise to find the maximum value of $|z|$??
. How could you find the exact value of the magnitude of the point highlighted??
Use Pythagoras to find the distance to the centre from the origin and then add the radius.
. How could you find the exact value of the magnitude of the point which is highlighted??
Use Pythagoras to find the distance to the centre from the origin and then subtract the radius. Make sure answer is positive.
The shortest distance from a point to a line always forms which angle??
$$ 90^{\circ} $$
If you have a line $y = mx + c$ and you wish to find the shortest distance from the origin to a point on that line, what would you do??
- Find the perpindicular gradient for that line
- $c$ is going through the origin
- Set the new equation equal to the original and solve for $x$ and $y$.
How could you rephrase the problem of finding the minimum value of $|z|$ for a perpindicular bisector??
Find the point of intersection between the perpindicular bisector and the line passing through the origin perpindicular to the perpindicular bisector.
If you know $\text{arg}((x - 2) + i(y - 2)) = \frac{\pi}{6}$ and $x + iy$ is in the first quadrant of the Argand Diagram, how could you begin to solve the problem??
$$ \tan\left(\frac{(y-2)}{(x-2)}\right) = \frac{\pi}{6} $$
What’s the best thing to do when attempting a loci question??
Draw an Argand Diagram.
What’s the best way to think about a complex loci question??
A normal co-ordinate geometry question in disguise.
Backlinks
- [[Further Maths - Complex Numbers]]S
- [[Further Maths - Regions in the Argand Diagram]]S
- [[Further Maths - Syllabus]]S
Metadata
date: 2020-11-11 18:46
tags:
- '@?further-maths'
- '@?complex-numbers'
- '@?public'
title: Further Maths - Loci in the Argand Diagram