# 2 - Coordinate Geometry

##### 2016-09-20 12:21:00 +0000, 2 years and 3 weeks ago

1 - Equation of a straight line

• The distance between two points
• Mid-point of a line
• Familiarity with straight line equations

To find the distance between two points, AB, shown:

|
|              B * (x_2, y_2)
|               /|
|              / |
|             /  |
|            /   |
|           /    |
|          /     :      o = point
|         /      |      C = x_A, y_B intercept, forms
|        /       |          a triangle at 90 degrees.
|       /        |
|      /         |
|   A /          |
|    *-----------* C
|   (x_1, y_1)
|
|
+---------------------


A is at x1, y1 and B is at x2, y2, so in order to find distance AB you must find the distance between AC and AB, according to pythagoras:

Therefore AB is equal to:

Now that we know the distance AB, we can find the mid point of it.
We call this midpoint ‘M’.

Half of AC is the x coordinate of M, and half of BC is the y coordinate, therefore:

As shown in the below diagram:

|
|              B * (x_2, y_2)
|               /|
|              / |
|             /  |
|            /   :
|           /    |
|          *-----+   ^  1/2(x_2, y_2)
|         /|     |   |
|        / |     |   |
|       /  :     :   |
|      /   |     |   |
|   A /    |     |   |
|    *-----+-----* C v
|   (x_1, y_1)
|    <----->
|   1/2(x_1, y_1)
|
+---------------------


The equation of a curve is a rule for determining whether of not the point with the co-ordinates (x,y) lies on the curve.

e.g. Eqn. of the line through (x,y) with gradient M.
M is equal to the y step / x step, what I mean by this is how far up the y axis goes up per x point.

|
|              B o (x_2, y_2)
|               /
|              /
|             /
|            /
|           /
|          /
|         /
|        /
|       /
|      /
|   A /
|    o
|   (x_1, y_1)
|
|
+---------------------


Therefore,

When in the form (x,y), you can use the below formula to find the gradient.

If two lines are parallel if they have the same gradient.
Perpendicular lines meet each other at 90 degrees.

Gradient of a perpendicular line is therefore:

When the line is paralell to the x axis, according to $y=mx+c$, the answer is always y = y intercept

  | y
|
|
|
-2+------------>
|
|
--+------------> x


Where ‘m’ is the gradient, in this case, 0, since it is parallel to x, therefore:

And c is the y intersect, 2, therefore $y=3$

| y |
|   |
|   |
|   |
|   |
|   |
|   |
+---+--------> x
3


In this case where y is paralell to the y axis, the answer will always be $x=c etc.$

Most questions are given in the form of:

And you are told to find the equation of the co-ordinates, we can use:

To find that line, however; we are not given a gradient of that line unlike before, but since the two lines are parallel, they therefore must have the same gradient. In $y=4x-1$, we are given the gradient, ‘4’ in this case, we can put that into the ‘m’ of our equation.

Perpendicular means 90 degrees to, to find the gradient of the perpendicular line, we must do the negative of the reciprocal.
Reciprocal means to invert $\frac{1}{x}$ so that $\frac{x}{1}$, we then negate it so that it is now $-\frac{x}{1}$, the general rule for negative fractions is:

|
| m   A_1
|\----+    /| B_1
| \   |   / |
|B2\  :i /  :
|   \ | /   | m
|    \|/    |
|     +-----+ A
|    / \  i
|   / P \
|  /     \
| /       \
|/         \
+-----------------


The gradient of the perpendicular line is therefore; $m_1 = \frac{-1}{m_1}$ Therefore the perpendicular y is now equal to:

3 - Coordinate geometry of a circle

4 - Equation of a circle

5 - Equation of a tangent and normal

6 - Intersection of straight lines & curves

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