AS Level
Mathematics 9709 P1

3. Coordinate Geometry

Written by: Tharun Athreya
Formatted by: Dhyaneshwaran V

Basic coordinates geometry refers to working with points, lines and shapes on the coordinate axis.
Using coordinates, you can:
- Calculate the distance between two points (length of a line)
- Find the equation of the tangents and normal.
- Find the midpoint of a line
- Divide lines in m:n ratio
- Calculate the area of a triangle
To find the midpoint: M = $$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
To find the length: $$\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$

Gradient of a line = $$\text{Gradient} = \frac{y_2 – y_1}{x_2 – x_1}$$
If two lines are in parallel, then their gradients are equal.
Perpendicular Lines: If a line has a gradient m, then every line perpendicular to it has gradient $$-\frac{1}{m}$$
- Rule for perpendicular lines: $$m_1 \times m_2 = -1$$

Equation of the straight line: y=mx+c
- $ m$ is the gradient
- $ c$ is the y-axis intercept
Alternative form is $ ax + by + c = 0 $
where $ a, b, \text{and } c $ are integers.
You could also use: $ y – y_1 = m(x – x_1) $
Collinear lines are the same straight line so the gradients are equal

A circle with centre (a, b) and the radius r has the equation:
$$(x – h)^2 + (y – k)^2 = r^2
$$
Expanding the equation $ (x – a)^2 + (y – b)^2 = r^2 $ gives:
$$x^2 – 2ax + a^2 + y^2 – 2by + b^2 = r^2
$$
Rearranging gives:
$$x^2 + y^2 – 2ax – 2by + (a^2 + b^2 – r^2) = 0$$
- The coefficient of $x^2$and $y^2$ are equal
- There is no $ xy $ term.

🔥 Key Point

While using the expanded form of a circle, there is a easier way to find the centre and the radius:

$ x^2 + y^2 + 2gx + 2fy + c = 0 $
- where $ (-g, -f) $ is the centre and $ \sqrt{g^2 + f^2 – c} $ is the radius

How can I use perpendicular bisectors to find the equation of a circle?
- A chord of a circle is a straight line segment between any two points on the circle.
- The perpendicular bisector of a chord always goes through the centre of the circle.
- If you know three points on a circle, draw any two chords between them – the perpendicular bisectors of the chords will meet at the centre of the circle.
The angle in a semicircle is a right angle.
- The angle in a semicircle property says that If a triangle is right-angled, then its hypotenuse is a diameter of its circumcircle.
How can I use the angle in a semicircle property to find the equation of a circle?
- Firstly the hypotenuse of a right-angled triangle is a diameter of the triangle’s circumcircle you also know that:
  - the radius of the circumcircle is half the length of the hypotenuse
  - the centre of the circumcircle is the midpoint of the hypotenuse
- Once you know the radius and the centre you can write down the equation of the circle
A tangent is the line which meets the circle at only one point but doesn’t cut across the circle.
- A tangent to a circle is perpendicular to the radius of the circle at the point of intersection