Coordinate geometry in the (x, y) plane

Mastering Coordinate Geometry: The Straight Line

Welcome to one of the most useful chapters in your AS Level Mathematics journey! Coordinate Geometry is the bridge between algebra and shapes. It allows us to take a line or a curve and describe it perfectly using an equation.

Why is this important? Without coordinate geometry, we wouldn't have GPS navigation, computer-aided design (CAD) for architecture, or even the graphics in your favorite video games! In this chapter, we will focus on the most fundamental shape of all: The Straight Line.

1. The Steepness: Gradient (\( m \))

Before we can build a line, we need to know its gradient (also called the slope). The gradient tells us how steep a line is and which direction it's going.

Imagine you are walking from left to right:
• If the line goes up, the gradient is positive.
• If the line goes down, the gradient is negative.
• If the line is flat, the gradient is zero.

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

Memory Aid: Think "Rise over Run." The "Rise" (change in \( y \)) goes on top, and the "Run" (change in \( x \)) goes on the bottom.

Common Mistake to Avoid: Always subtract your coordinates in the same order! If you start with \( y_2 \) on the top, you must start with \( x_2 \) on the bottom.

Key Takeaway:

The gradient \( m \) measures the "steepness." Larger numbers mean steeper lines!

2. Equations of a Straight Line

The syllabus requires you to know two main ways to write the equation of a line. Don't worry if these look different; they all describe the same thing!

Form A: The Point-Gradient Form
\( y - y_1 = m(x - x_1) \)
This is the "secret weapon" for exam students. It is the easiest way to find the equation of a line if you know the gradient (\( m \)) and any single point \( (x_1, y_1) \) on that line.

Form B: The General Form
\( ax + by + c = 0 \)
In this form, \( a \), \( b \), and \( c \) are usually whole numbers (integers). You will often be asked to "give your answer in the form \( ax + by + c = 0 \)."

Step-by-Step: Finding a line through two points
If you are given points \( A(2, 5) \) and \( B(4, 9) \):
1. Find the gradient: \( m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2 \).
2. Pick a point: Let's use \( A(2, 5) \).
3. Plug into the formula: \( y - 5 = 2(x - 2) \).
4. Simplify: \( y - 5 = 2x - 4 \).
5. Rearrange (if needed): \( y = 2x + 1 \).

Key Takeaway:

Always try to use \( y - y_1 = m(x - x_1) \) first. It’s faster and leads to fewer mistakes than the old \( y = mx + c \) method!

3. Parallel and Perpendicular Lines

Lines have special relationships based on their gradients. This is a very common exam topic!

Parallel Lines:
These are like train tracks—they never meet. Because they have the same steepness, their gradients are equal.
If line 1 has gradient \( m_1 \) and line 2 has gradient \( m_2 \), then:
\( m_1 = m_2 \)

Perpendicular Lines:
These lines meet at a perfect 90-degree angle (like a cross). Their gradients are the "negative reciprocal" of each other.
The rule is:
\( m_1 \times m_2 = -1 \)

A Simple Trick: To find a perpendicular gradient, "Flip it and Switch it!"
• If your gradient is \( \frac{2}{3} \), flip the fraction to \( \frac{3}{2} \) and switch the sign to negative: \( -\frac{3}{2} \).
• If your gradient is \( 4 \) (which is \( \frac{4}{1} \)), flip it to \( \frac{1}{4} \) and switch the sign: \( -\frac{1}{4} \).

Did you know? The word "perpendicular" comes from the Latin perpendiculum, which means "plumb line"—a tool builders used to make sure walls were perfectly straight up and down!

Key Takeaway:

Parallel = Same gradient.
Perpendicular = Gradients multiply to make \( -1 \).

4. Putting it All Together: Exam-Style Example

Question: Find the equation of the line that passes through the point (2, 3) and is perpendicular to the line with equation \( 3x + 4y = 18 \).

Step 1: Find the gradient of the given line.
Rearrange \( 3x + 4y = 18 \) into the form \( y = mx + c \):
\( 4y = -3x + 18 \)
\( y = -\frac{3}{4}x + \frac{18}{4} \)
The gradient of this line is \( -\frac{3}{4} \).

Step 2: Find the perpendicular gradient.
Using "Flip it and Switch it," the perpendicular gradient \( m \) is \( \frac{4}{3} \).

Step 3: Use the point-gradient formula.
We have the point (2, 3) and the gradient \( m = \frac{4}{3} \).
\( y - 3 = \frac{4}{3}(x - 2) \)

Step 4: Clean it up.
Multiply everything by 3 to get rid of the fraction:
\( 3(y - 3) = 4(x - 2) \)
\( 3y - 9 = 4x - 8 \)
Final answer (in general form): \( 4x - 3y + 1 = 0 \).

Quick Review Checklist

• Do I know the gradient formula? \( \frac{y_2 - y_1}{x_2 - x_1} \)
• Can I use the \( y - y_1 = m(x - x_1) \) formula?
• Do I remember that parallel lines have the same gradient?
• Can I find a perpendicular gradient by flipping and changing the sign?
• Can I rearrange an equation into the \( ax + by + c = 0 \) form?

Final Encouragement: Coordinate geometry can feel like a lot of symbols at first, but it is very logical. Once you master finding the gradient and using the point-gradient formula, you can solve almost any problem in this chapter. Keep practicing!