# Further Maths - Loci in the Argand Diagram

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/loci-argand-diagram/ · Updated: 2024-08-24 · Tags: further-maths, complex-numbers

### Distance between complex numbers
##### In ![PHOTO](argand-diagram-dist-two-points.png), how could you write the distance between the two points??
$$
|z_2 - z_1|
$$
^distance-between-two-points-modulus

##### ![PHOTO](argand-diagram-dist-two-points-completed.png) This result is a case of what result for vectors??
$$
\vec{AB} = b - a
$$
^modulus-distance-as-vector-result

##### For two complex numbers $z_1$ and $z_2$, how could you write the distance between them??
$$
|z_2 - z_1|
$$
^distance-between-z1-z2-modulus

##### 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 $z_2$.
^variable-z-distance-to-fixed-point

##### What is |z_2 - z_1| for two complex numbers??
The distance between $z_2$ and $z_1$.
^modulus-difference-is-distance

### Circles from a fixed modulus
##### 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 $r$.
^modulus-equals-r-is-circle

##### If $z_1 = a+bi$ and $|z-z_1| = r$, what is the radius of the circle formed??
$$
r
$$
^circle-radius-from-modulus-equation

##### If $z_1 = a+bi$ and $|z-z_1| = r$, where is the centre of the circle??
$$
(a, b)
$$
^circle-centre-from-modulus-equation

##### How could you rewrite $|z - 4 + 8i|$ more usefully??
$$
|z - (4 - 8i)|
$$
^rewrite-modulus-as-z-minus-point

##### Where is the centre of $|z - (1 + 2i)| = 4$??
$$
(1, 2)
$$
^circle-centre-bracketed-example

##### What is the radius of $|z - (1 + 2i)| = 4$??
$$
4
$$
^circle-radius-bracketed-example

##### Where is the centre of $|z - 1 + 2i| = 4$??
$$
(1, -2)
$$
^circle-centre-unbracketed-example

### Cartesian equation of a circle
##### If $z = x + yi$ and $z_1 = x_1 + y_1 i$, how could you rewrite $|z - z_1|$??
$$
| (x - x-1) + i(y - y1) |
$$
^modulus-in-cartesian-components

##### What do you get if you square both sides of $| (x - x-1) + (y - y1) | = r$??
$$
(x - x-1)^2 + (y - y1)^2 = r^2
$$
^square-modulus-to-cartesian-circle

##### What is $(x - x-1)^2 + (y - y1)^2 = r^2$??
The Cartesian equation of a circle.
^cartesian-equation-of-circle

### Recognising loci from diagrams
##### ![PHOTO](argand-diagram-circle.png) describes what relationship??
$$
|z - z_1| = r
$$
^diagram-shows-circle-relationship

##### ![PHOTO](argand-diagram-perp-bisector.png) describes what relationship??
$$
|z - z_1| = |z - z_2|
$$
^diagram-shows-perpendicular-bisector-relationship

##### What is the locus of points that are an equal distance from two different points $z_1$ and $z_2$??
The perpendicular bisector of the line segment joining the two points.
^equidistant-locus-is-perpendicular-bisector

##### What is $|z - z_1| = |z - z_2|$??
The locus of points forming a perpendicular bisector of the line segment joining $z_1$ and $z_2$.
^equal-moduli-is-perpendicular-bisector

### Intersecting a circle with the imaginary axis
##### What is the equation of the following circle ![PHOTO](argand-diagram-circle-real.png)??
$$
|z - (5 + 4i)| = 8
$$
^circle-equation-from-diagram-example

##### What are the points in ![PHOTO](argand-diagram-circle-real.png) where $\text{Re}(z) = 0$??
![PHOTO](argand-diagram-circle-real-label.png)
^circle-points-on-imaginary-axis

##### What lines could you visualise in ![PHOTO](argand-diagram-circle-real-label.png) in order to find the points of intersection with the imaginary axis??
![PHOTO](argand-diagram-circle-real-tri.png)
^visualise-triangle-for-axis-intersection

##### ![PHOTO](argand-diagram-circle-real-tri.png) What are the lengths of the line segments??
![PHOTO](argand-diagram-circle-real-tri-label.png)
^triangle-side-lengths-labelled

##### ![PHOTO](argand-diagram-circle-real-tri-label.png) What is the length of the unlabeled side length??
$$
\sqrt{8^2 - 5^2}
$$
^unlabelled-side-by-pythagoras-value

##### ![PHOTO](argand-diagram-circle-real-tri-label.png) How could you find the length of the unlabelled side??
Using Pythagoras.
^unlabelled-side-method-pythagoras

##### ![PHOTO](argand-diagram-circle-real-tri-label.png) If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the top point??
$$
4 + \sqrt{8^2 - 5^2}
$$
^top-intersection-vertical-position

##### ![PHOTO](argand-diagram-circle-real-tri-label.png) If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the bottom point??
$$
4 - \sqrt{8^2 - 5^2}
$$
^bottom-intersection-vertical-position

### Cartesian equation of a perpendicular bisector
##### What does the Cartesian equation for a perpendicular bisector look like??
$$
y = mx + c
$$
^perpendicular-bisector-cartesian-form

##### How can you find the Cartesian equation for a perpendicular bisector??
* Find the equation of the lines passing through two points
* Find the perpendicular line at the midpoint
^find-perpendicular-bisector-cartesian-method

### Maximum and minimum modulus on a circle
##### ![PHOTO](argand-diagram-circle-2.png) This the relationship $|z - (a + bi)| = r|$. What line could you visualise to find the maximum value of $|z|$??
![PHOTO](argand-diagram-circle-2-max-mod.png)
^visualise-line-for-max-modulus-on-circle

##### ![PHOTO](argand-diagram-circle-2-max-mod.png). 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.
^max-modulus-distance-plus-radius

##### ![PHOTO](argand-diagram-circle-2-min-mod.png). 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.
^min-modulus-distance-minus-radius

### Shortest distance and minimum modulus on a line
##### The shortest distance from a point to a line always forms which angle??
$$
90^{\circ}
$$
^shortest-distance-forms-right-angle

##### 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 perpendicular gradient for that line
* $c$ is going through the origin
* Set the new equation equal to the original and solve for $x$ and $y$.
^shortest-distance-origin-to-line-method

##### How could you rephrase the problem of finding the minimum value of $|z|$ for a perpendicular bisector??
Find the point of intersection between the perpendicular bisector and the line passing through the origin perpendicular to the perpendicular bisector.
^min-modulus-on-bisector-as-intersection

### Argument loci and general advice
##### 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??
$$
\tan\left(\frac{(y-2)}{(x-2)}\right) = \frac{\pi}{6}
$$
^argument-locus-tangent-setup

##### What's the best thing to do when attempting a loci question??
Draw an Argand Diagram.
^loci-question-draw-argand-diagram

##### What's the best way to think about a complex loci question??
A normal co-ordinate geometry question in disguise.
^complex-loci-as-coordinate-geometry

##### ![PHOTO CIRCLE LOCI QUICK](circle-loci-quick.png) What method do I often forget that is a much quicker way to find the minimum and maximum $|z|$??
Use the distance to the center and then add on the radius.
^quick-min-max-modulus-distance-radius

### Areas in loci questions
##### ![PHOTO WEIRD CIRCLE REGION](weird-circle-region.png) What technique could be used to find the area here??
Polar integration.
^area-of-loci-region-polar-integration

##### What technique can sometimes be used to make finding areas in loci questions much easier??
Polar integration.
^loci-areas-polar-integration

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