Further Maths - Loci in the Argand Diagram

840adf4c175e4b1780f283b85f696e3d

In , how could you write the distance between the two points?

\[\vert z _ 2 - z _ 1 \vert\]

affa0a6020be4e62a534e90403230365

This result is a case of what result for vectors?

\[\vec{AB} = b - a\]

0daf09d4d158421d963819724275a3de

For two complex numbers $z _ 1$ and $z _ 2$, how could you write the distance between them?

\[\vert z _ 2 - z _ 1 \vert\]

b589020b5e1a4b94ada7d61b21eee63a

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 $z _ 2$.

0babfc5d26f74342ab5d0f79383b6099

What is

z _ 2 - z _ 1

for two complex numbers?

The distance between $z _ 2$ and $z _ 1$.

18e19c8a6a6045d49571fd43d7f35417

The equation $ \vert z - z _ 1 \vert = r$ means what in practical terms?

The distance between $z _ 1$ and $z$ is fixed at $r$. This describes a circle with radius $r$.

e3cba9f79a834270990e28f1437a7321

If $z _ 1 = a+bi$ and $ \vert z-z _ 1 \vert = r$, what is the radius of the circle formed?

\[r\]

0ad903a596c54f678818440c87cf1815

If $z _ 1 = a+bi$ and $ \vert z-z _ 1 \vert = r$, where is the centre of the circle?

\[(a, b)\]

1b15c418cc594c61903f844afce1351a

Where is the centre of $ \vert z - (1 + 2i) \vert = 4$?

\[(1, 2)\]

2f40d585344548e59c5557dc6e5409ba

What is the radius of $ \vert z - (1 + 2i) \vert = 4$?

\[4\]

b613974de9714c8fb3323f451c3f5c09

Where is the centre of $ \vert z - 1 + 2i \vert = 4$?

\[(1, -2)\]

a566d45c24fc4817bcedc95e9e35ce46

If $z = x + yi$ and $z _ 1 = x _ 1 + y _ 1 i$, how could you rewrite $ \vert z - z _ 1 \vert $?

\[\vert (x - x-1) + i(y - y1) \vert\]

2d02968f2953430287223fc856af0136

What do you get if you square both sides of $ \vert (x - x-1) + (y - y1) \vert = r$?

\[(x - x-1)^2 + (y - y1)^2 = r^2\]

b529886d9e7147c3a57433755541c77a

What is $(x - x-1)^2 + (y - y1)^2 = r^2$?

The Cartesian equation of a circle.

f3641f3a9b944202925aaa3b712b6031

describes what relationship?

\[\vert z - z _ 1 \vert = r\]

efac0e85e0a4450290dc6735cfa5b46a

describes what relationship?

\[\vert z - z _ 1 \vert = \vert z - z _ 2 \vert\]

12f72cd171ad43309616a136c589e1de

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.

779add0e973d4442adcd343464525546

What is $ \vert z - z _ 1 \vert = \vert z - z _ 2 \vert $?

The locus of points forming a perpindicular bisector of the line segment joining $z _ 1$ and $z _ 2$.

a1c4aba7e59c4b50ae0a78ab166cddd1

What is the equation of the following circle ?

\[\vert z - (5 + 4i) \vert = 8\]

bd5c00668e24438bb348e917e60d9299

What are the points in where $\text{Re}(z) = 0$?

bc035114a075433c8761811ba09e4533

What lines could you visualise in in order to find the points of intersection with the imaginary axis?

f76ff7992fd34899850253b9ba2fd1f6

What are the lengths of the line segments?

02e4bf244c0f4af8bc6f1dcc2db0b80b

What is the length of the unlabeled side length?

\[\sqrt{8^2 - 5^2}\]

56b7e0db4d52486388da4dde910439dc

How could you find the length of the unlabelled side?

Using Pythagoras.

834a36bcf703416c843601933afb9fdb

If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the top point?

\[4 + \sqrt{8^2 - 5^2}\]

49de476386284a8bb729a013988bf03d

If the unlabeled side length is $\sqrt{8^2 - 5^2}$, what is the vertical position of the bottom point?

\[4 - \sqrt{8^2 - 5^2}\]

a4d22f8e8ef14b5eb283f58826427047

What does the Cartesian equation for a perpindicular bisector look like?

\[y = mx + c\]

2527686a7a314e958d930b91f0a65e23

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

1a73e09149c74163a0f6289e45e9faec

This the relationship $ \vert z - (a + bi) \vert = r \vert $. What line could you visualise to find the maximum value of $ \vert z \vert $?

d842b69df3954670bfc14f58e003f603

. 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.

89aee1148e1c478da79b30ac831567dc

. 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.

b26be7d2a56a49d681e4115d0c7a39bb

The shortest distance from a point to a line always forms which angle?

\[90^{\circ}\]

d294e29bed5a4f54a2110dbb17972946

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 $y$.

3df4b14120ef4ff0919500d07b8bb1cb

How could you rephrase the problem of finding the minimum value of $ \vert z \vert $ 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.

623f1b0eb8c544ef984f6a3b0df8780a

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?

\[\tan\left(\frac{(y-2)}{(x-2)}\right) = \frac{\pi}{6}\]

1d1c03ad0f804c9db334232e8c474fbd

What’s the best thing to do when attempting a loci question?

Draw an Argand Diagram.

7e8a8de937f440b9a4a8e4247820b5e8

What’s the best way to think about a complex loci question?

A normal co-ordinate geometry question in disguise.

64ce0a8bb02340c286188986e4f4a39e

PHOTO CIRCLE LOCI QUICK What method do I often forget that is a much quicker way to find the minimum and maximum $ \vert z \vert $?