Coordinate geometry in the (x, y) plane

Welcome to Coordinate Geometry!

In this chapter of Pure Mathematics 1 (P1), we are going to explore how to describe straight lines using algebra. Think of coordinate geometry as a bridge between shapes (geometry) and equations (algebra). By the end of these notes, you’ll be able to map out any straight line on a graph just by looking at its "DNA"—its equation!

Why does this matter? Architects use these principles to design buildings, and computer programmers use them to create graphics in your favorite video games. If you can master the straight line, you’re well on your way to mastering calculus later on. Don't worry if this seems a bit abstract at first; we’ll take it one step at a time!

1. The Foundation: The Gradient (m)

Before we build an equation, we need to understand the gradient. The gradient tells us two things: how steep a line is and which direction it’s going.

The Concept: Imagine you are walking up a hill. The gradient is simply the "vertical rise" divided by the "horizontal run."

The Formula: If you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the gradient \( m \) is calculated as:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Quick Tips:
• If the line goes up from left to right, the gradient is positive.
• If the line goes down from left to right, the gradient is negative.
• A horizontal line has a gradient of 0.
• A vertical line has an undefined gradient.

Did you know? The letter \( m \) is used for gradient, possibly from the French word monter, which means "to climb"!

2. The Equation of a Straight Line

The syllabus requires you to be comfortable with two main forms of a straight-line equation. Mastering these will make exam questions much easier to navigate.

Form A: The Point-Slope Form

This is often the most useful version for P1 students. You use this when you know one point on the line \( (x_1, y_1) \) and the gradient \( m \).
The Formula: \( y - y_1 = m(x - x_1) \)

Step-by-Step Example: Find the equation of a line with gradient 4 passing through the point \( (2, 3) \).
1. Identify your values: \( m = 4 \), \( x_1 = 2 \), and \( y_1 = 3 \).
2. Plug them into the formula: \( y - 3 = 4(x - 2) \).
3. Expand and simplify: \( y - 3 = 4x - 8 \), so \( y = 4x - 5 \).
Easy, right? You just "plug and play"!

Form B: The General Form

Sometimes, the exam will ask you to give your answer in the form \( ax + by + c = 0 \), where \( a, b, \) and \( c \) are integers (whole numbers).
Example: If you have \( y = \frac{2}{3}x + 5 \), you can convert it by multiplying everything by 3 to get rid of the fraction: \( 3y = 2x + 15 \). Then move everything to one side: \( 2x - 3y + 15 = 0 \).

Key Takeaway:

Always read the question carefully to see which form the examiner wants. If they don't specify, \( y - y_1 = m(x - x_1) \) is your best friend!

3. Parallel and Perpendicular Lines

This is a favorite topic for examiners! It's all about how the gradients of two different lines relate to each other.

Parallel Lines

Parallel lines are like train tracks—they never meet because they are tilted at the exact same angle. Therefore, their gradients are equal.
Condition: \( m_1 = m_2 \)

Perpendicular Lines

Perpendicular lines meet at a perfect right angle (\( 90^\circ \)). Their gradients have a very specific relationship: when you multiply them together, you get -1.
Condition: \( m_1 \times m_2 = -1 \)

The Memory Trick: "Flip and Switch"
To find a perpendicular gradient, take the original gradient, flip it upside down (the reciprocal), and switch the sign (+ to - or - to +).
Example: If the gradient of Line A is \( 3 \), the perpendicular gradient of Line B is \( -\frac{1}{3} \).
Example: If the gradient of Line A is \( -\frac{2}{5} \), the perpendicular gradient of Line B is \( \frac{5}{2} \).

Common Mistake to Avoid: Students often forget to change the sign. Remember, if one line is going up, the line at a right angle to it must be going down!

4. Finding the Equation through Two Points

What if you aren't given the gradient? What if you only have two points, like \( A(1, 2) \) and \( B(3, 10) \)?
Don't panic! Just follow these two steps:
1. Find the gradient (\( m \)) using the two-point formula: \( m = \frac{10 - 2}{3 - 1} = \frac{8}{2} = 4 \).
2. Pick ONE point (either one works!) and use the point-slope formula: \( y - 2 = 4(x - 1) \).
3. Simplify: \( y = 4x - 2 \).

5. Quick Review & Tips for Success

Quick Review Box:
• Gradient: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Line Equation: \( y - y_1 = m(x - x_1) \)
• Parallel: \( m_1 = m_2 \)
• Perpendicular: \( m_1 = -\frac{1}{m_2} \)

Exam Pro-Tips:
• Sketch it out: If you're stuck, draw a quick sketch of the coordinates. It helps you visualize if your gradient should be positive or negative.
• Fraction Action: Examiners love using fractions. Keep your gradients as fractions (like \( \frac{2}{3} \)) rather than decimals (like \( 0.666... \)) for better accuracy.
• Watch your signs: Subtracting a negative number is the same as adding! \( y - (-3) \) becomes \( y + 3 \). This is where most marks are lost!

Summary:

Coordinate geometry in P1 is all about understanding the relationship between the steepness (gradient) and the position (points) of a line. Once you can find the gradient and use the \( y - y_1 = m(x - x_1) \) formula, you can solve almost any problem in this chapter! Keep practicing, and it will become second nature.