Coordinate geometry

Welcome to Coordinate Geometry!

In this chapter, we are going to learn how to describe shapes, lines, and positions using numbers and algebra. Think of coordinate geometry as a bridge between algebra and shapes. By the end of these notes, you’ll be able to track paths of straight lines and define the perfect circle using nothing but equations!

Why is this important? Coordinate geometry is used everywhere—from GPS navigation in your phone to architects designing curved buildings and video game developers calculating how a character moves across the screen.

1. Straight Lines (Section C1)

A straight line is the simplest path between two points. In AS Level, we move beyond the basic \( y = mx + c \) you might remember from GCSE and look at more professional ways to write these equations.

The Two Main Forms

1. The Point-Gradient Form: \( y - y_1 = m(x - x_1) \)
This is your new best friend. Use this when you know one point on the line \( (x_1, y_1) \) and the gradient \( m \). It is much faster than solving for \( c \)!

2. The General Form: \( ax + by + c = 0 \)
In this form, \( a \), \( b \), and \( c \) are usually integers (whole numbers). This is a very neat way to present your final answer.

Parallel and Perpendicular Lines

Understanding how lines interact is a huge part of Paper 1.

• Parallel Lines: These lines have the same gradient. If line A has a gradient of \( 3 \), any line parallel to it also has a gradient of \( 3 \).
• Perpendicular Lines: These lines meet at a right angle (\( 90^\circ \)). Their gradients are negative reciprocals of each other. This means if you multiply them together, you get \( -1 \).

Memory Aid: The "Flip and Swap" Trick
To find a perpendicular gradient: 1. Flip the fraction (e.g., \( 2 \) becomes \( \frac{1}{2} \)). 2. Swap the sign (e.g., positive becomes negative). Example: If the gradient is \( \frac{3}{4} \), the perpendicular gradient is \( -\frac{4}{3} \).

Common Mistake: When using \( ax + by + c = 0 \), many students forget that the gradient is actually \( -\frac{a}{b} \). Always double-check your signs!

Quick Review: - Use \( y - y_1 = m(x - x_1) \) for quick calculations. - Parallel: \( m_1 = m_2 \). - Perpendicular: \( m_1 \times m_2 = -1 \).

2. The Geometry of the Circle (Section C2)

A circle is defined by two things: where its centre is and how big its radius is.

The Standard Equation

The equation of a circle with centre \( (a, b) \) and radius \( r \) is:
\( (x - a)^2 + (y - b)^2 = r^2 \)

Did you know? This equation is actually just Pythagoras' Theorem in disguise! It measures the distance from the centre to any point on the edge.

Finding the Centre and Radius

Sometimes the exam will give you a "messy" equation like \( x^2 + y^2 - 4x + 6y - 12 = 0 \). To find the centre and radius, you need to complete the square for both \( x \) and \( y \).

Step-by-Step Process: 1. Group the \( x \) terms and \( y \) terms together. 2. Complete the square for the \( x \) parts. 3. Complete the square for the \( y \) parts. 4. Move all the constant numbers to the right side of the equals sign. 5. The right side is \( r^2 \), so square root it to find the radius!

Key Circle Properties (Theorems)

The syllabus requires you to use these three geometric rules in your coordinate geometry problems:

• Angle in a semicircle: If you draw a triangle from the ends of the diameter to any point on the circumference, the angle is always \( 90^\circ \).
• Perpendicular bisector of a chord: A line drawn from the centre of the circle that hits a chord at \( 90^\circ \) will always cut that chord exactly in half.
• Tangents: A tangent is a line that just touches the edge of the circle. The radius at that point and the tangent are always perpendicular (\( 90^\circ \)).

Don't worry if this seems tricky at first! Just remember that most circle questions are just hidden "gradient" questions. If you know the radius and tangent are perpendicular, you can use the "Flip and Swap" trick we learned earlier.

Key Takeaway: - Circle Equation: \( (x - a)^2 + (y - b)^2 = r^2 \). - Centre is \( (a, b) \)—watch out for the signs! - Radius is \( \sqrt{r^2} \).

3. Modeling with Straight Lines

Sometimes, math isn't just about \( x \) and \( y \); it’s about real life. You might be asked to use a straight line to model things like cost vs time.

Example: A taxi charges a fixed "pick-up" fee plus a cost per mile. - The gradient (\( m \)) represents the cost per mile. - The y-intercept (\( c \)) represents the fixed pick-up fee.

Analogy: Imagine climbing a hill. The "gradient" is how steep the hill is, and the "y-intercept" is the height where you started climbing.

Common Mistake: In modeling questions, always check your units. If the gradient is in "pence per mile" but the intercept is in "pounds," your equation won't work!

Summary Checklist

Before moving on to the next chapter, make sure you can:
- Write the equation of a line using \( y - y_1 = m(x - x_1) \).
- Find a perpendicular gradient by "flipping and swapping."
- Identify the centre and radius of a circle from its equation.
- Complete the square to fix "messy" circle equations.
- Use the fact that a tangent is perpendicular to the radius to solve problems.