Coordinate geometry in the (x,y) plane

Introduction to Coordinate Geometry

Welcome to the world of Coordinate Geometry! In this chapter, we bridge the gap between algebra and shapes. By using an \((x, y)\) grid, we can describe lines, circles, and complex curves using equations. This is a vital skill for your A Level journey, as it connects to everything from mechanics to computer graphics. Don't worry if it seems a bit abstract at first—once you see the patterns, it becomes one of the most rewarding parts of Pure Mathematics.

3.1 The Straight Line

A straight line is the simplest path between two points. At A Level, we move beyond the basic \(y = mx + c\) to more flexible forms.

Key Equations of a Line

1. The Point-Gradient Form: \(y - y_1 = m(x - x_1)\)
This is your "best friend" in the exam. If you have a gradient (\(m\)) and any point \((x_1, y_1)\), you can write the equation instantly without having to solve for \(c\).

2. The General Form: \(ax + by + c = 0\)
In this form, \(a, b,\) and \(c\) are usually integers. It’s a very tidy way to present your final answer.

Parallel and Perpendicular Lines

Comparing the steepness (gradients) of two lines tells us how they relate to each other:

• Parallel Lines: They have the same gradient. If line 1 has gradient \(m\), line 2 also has gradient \(m\).
• Perpendicular Lines: They meet at a 90° angle. Their gradients are negative reciprocals of each other. This means \(m' = -\frac{1}{m}\) or \(m \times m' = -1\).

Quick Review Box: To find the perpendicular gradient, "flip the fraction and change the sign." For example, if the gradient is \(2\), the perpendicular gradient is \(-\frac{1}{2}\). If it is \(-\frac{3}{4}\), the perpendicular gradient is \(\frac{4}{3}\).

Real-World Connection

Example: Straight line models are used to convert units (like Celsius to Fahrenheit) or to track distance over time for an object moving at a constant speed.

Key Takeaway: Always try to use \(y - y_1 = m(x - x_1)\) first—it's faster and reduces the chance of making a "sign error" when calculating the intercept.

3.2 Coordinate Geometry of the Circle

A circle is defined as all points that are a fixed distance (the radius) from a central point.

The Equation of a Circle

The standard form is: \((x - a)^2 + (y - b)^2 = r^2\)
• The centre is at \((a, b)\).
• The radius is \(r\).

Common Mistake Alert: Students often forget that the equation uses \(r^2\). If the equation ends in \(= 25\), the radius is \(5\), not \(25\)! Also, be careful with signs: \((x + 3)^2\) means the \(x\)-coordinate of the centre is \(-3\).

The Expanded Form

Sometimes you will see: \(x^2 + y^2 + 2fx + 2gy + c = 0\).
To find the centre and radius from this form, you must complete the square for both the \(x\) terms and the \(y\) terms.

Geometric Properties of Circles

These classic geometry rules are often tested using coordinates:

• Angle in a semicircle: The angle subtended by a diameter is always 90°.
• Chord Bisector: A line from the centre that is perpendicular to a chord will always cut that chord exactly in half (bisect it).
• Tangents: A tangent is a line that touches the circle at exactly one point. The radius drawn to that point is always perpendicular to the tangent.

Step-by-Step: Finding the Equation of a Tangent
1. Find the gradient of the radius (the line between the centre and the point of contact).
2. Calculate the perpendicular gradient (the negative reciprocal). This is the gradient of your tangent.
3. Use \(y - y_1 = m(x - x_1)\) with the point of contact and your new gradient.

Key Takeaway: Completing the square is the essential tool for turning a messy circle equation into its useful centre-radius form.

3.3 Parametric Equations

Sometimes it is easier to describe \(x\) and \(y\) separately using a third variable, usually \(t\) or \(\theta\). This third variable is called a parameter.

What is a Parameter?

Imagine a person walking along a path. Their \(x\) position and \(y\) position both change depending on the time (\(t\)).
Example: \(x = 2t\), \(y = t^2\).

Converting to Cartesian Form

To get back to a standard \(y = f(x)\) equation (the Cartesian form), you need to "eliminate the parameter."
1. Rearrange one equation to make \(t\) the subject (usually the simplest one).
2. Substitute this expression for \(t\) into the other equation.

Parametric Circles

Circles are often described using trigonometry:
\(x = a \cos t\)
\(y = a \sin t\)
This describes a circle with centre \((0,0)\) and radius \(a\). To turn this into a Cartesian equation, we use the identity \(\sin^2 t + \cos^2 t = 1\).

Did you know? Parametric equations are used in video game programming to move characters along specific curved paths or to calculate the trajectory of a projectile.

Key Takeaway: Parametric equations give you more control over specific sections of a curve by limiting the values of the parameter \(t\).

3.4 Parametric Modelling

In real-world contexts, parametric equations help us model motion. For example, if an object moves from point \(A\) to point \(B\) over a specific time, its position at any moment can be described parametrically.

Example: Constant Velocity

If an object moves from \((1, 8)\) at \(t=0\) to \((6, 20)\) at \(t=5\), we can find the equations for \(x\) and \(y\).
The change in \(x\) is \(5\) units over \(5\) seconds, so \(x = 1 + t\).
The change in \(y\) is \(12\) units over \(5\) seconds, so \(y = 8 + 2.4t\).

Key Takeaway: In modelling, always check the domain of the parameter (the range of values \(t\) can take) to ensure the model makes sense for the given context.

Final Chapter Summary

• Lines: Use \(y - y_1 = m(x - x_1)\) and remember the perpendicular gradient rule: \(m_1 m_2 = -1\).
• Circles: Know the form \((x - a)^2 + (y - b)^2 = r^2\). Use completing the square to find the centre and radius.
• Tangents: Remember the radius and tangent are perpendicular.
• Parametric: Eliminate the parameter to find the Cartesian equation, or use trig identities for circles.
• Modelling: Apply these equations to real-life paths and motion.

Don't worry if this seems tricky at first! Sketching a diagram is often the secret to seeing the solution in coordinate geometry. Keep practicing, and the patterns will reveal themselves!