Coordinate geometry in the (x, y) plane

Welcome to Coordinate Geometry!

Welcome to one of the most exciting parts of Mathematics! Coordinate Geometry is like a bridge between the world of shapes (geometry) and the world of numbers (algebra). By using an \(x\) and \(y\) grid, we can describe exactly where things are and how they move using simple equations.

Whether you are aiming for top marks or just trying to get through the basics, these notes will help you master the "language of the plane." Don't worry if it seems a bit abstract at first—once you see the patterns, it becomes much easier!

1. The Basics: Gradients, Midpoints, and Distance

Before we build houses (equations), we need our tools. These three formulas are your "toolbox" for any straight-line problem.

The Gradient (Slope)

The gradient, usually called \(m\), tells us how steep a line is. Think of it as "Rise over Run."

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Analogy: Imagine you are walking up a hill. If for every 1 meter you move forward (run), you go up 2 meters (rise), your gradient is 2. If you go down, the gradient is negative!

The Midpoint

The midpoint is the exact center between two points \((x_1, y_1)\) and \((x_2, y_2)\). It is simply the average of the \(x\)-coordinates and the average of the \(y\)-coordinates.

\(Midpoint = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)

The Distance Between Two Points

To find how long a line segment is, we use a formula based on Pythagoras' Theorem.

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Quick Review: Always keep your coordinates in the same order. If you start with point 2 for the \(y\)-values, start with point 2 for the \(x\)-values too!

2. Equations of a Straight Line

There are three main ways to write the equation of a line. Each one has a different "superpower."

Form 1: The Gradient-Intercept Form

\(y = mx + c\)

Here, \(m\) is the gradient and \(c\) is the y-intercept (where the line crosses the vertical axis).

Form 2: The Point-Gradient Form (The Student Favorite!)

\(y - y_1 = m(x - x_1)\)

This is the most useful version for exams. If you have any point \((x_1, y_1)\) and the gradient \(m\), you just plug them in. No need to solve for \(c\) immediately!

Form 3: The General Form

\(ax + by + c = 0\)

In this form, \(a, b,\) and \(c\) are usually integers. You will often be asked to "give your answer in the form \(ax + by + c = 0\)." To do this, just move everything to one side of the equals sign.

Key Takeaway: To find the equation of any straight line, you only need two things: a point and a gradient.

3. Parallel and Perpendicular Lines

Lines have relationships too! We can tell if lines are parallel or perpendicular just by looking at their gradients.

Parallel Lines

Parallel lines never meet because they have the exact same steepness. If line 1 has gradient \(m_1\) and line 2 has gradient \(m_2\), then:
\(m_1 = m_2\)

Perpendicular Lines

Perpendicular lines meet at a right angle (\(90^\circ\)). Their gradients are "negative reciprocals" of each other. The rule is:
\(m_1 \times m_2 = -1\)

Simple Trick: To find a perpendicular gradient, flip the fraction upside down and change the sign. For example, if a line has a gradient of \(\frac{2}{3}\), the perpendicular line has a gradient of \(-\frac{3}{2}\).

4. The Geometry of the Circle

A circle is defined by its centre and its radius. In the \((x, y)\) plane, we use a specific equation to describe it.

The Equation of a Circle

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

Common Mistake: Be careful with the signs! If the equation is \((x + 3)^2 + (y - 5)^2 = 16\), the centre is actually \((-3, 5)\) and the radius is \(\sqrt{16} = 4\).

Completing the Square

Sometimes the exam will give you an expanded equation like \(x^2 + 4x + y^2 - 6y - 12 = 0\). To find the centre and radius, you must complete the square for both \(x\) and \(y\).
1. Group the \(x\)'s: \((x^2 + 4x) \rightarrow (x + 2)^2 - 4\)
2. Group the \(y\)'s: \((y^2 - 6y) \rightarrow (y - 3)^2 - 9\)
3. Put it back together: \((x + 2)^2 - 4 + (y - 3)^2 - 9 - 12 = 0\)
4. Simplify: \((x + 2)^2 + (y - 3)^2 = 25\)
Now we can see the centre is \((-2, 3)\) and the radius is \(5\).

Did you know? Translating a circle (moving it) just changes the \((a, b)\) values in the equation. The radius stays exactly the same!

5. Tangents and Normals to Circles

A tangent is a straight line that touches the circle at exactly one point. A normal is a line that goes through the centre of the circle and is perpendicular to the tangent at that point.

Key Properties to Remember:

1. Radius-Tangent Rule: The tangent is always perpendicular to the radius at the point of contact.
2. Semicircle Rule: The angle in a semicircle is always a right angle (\(90^\circ\)).
3. Chord Rule: The perpendicular line from the centre to a chord always bisects (cuts in half) that chord.

Step-by-Step for Tangent Equations:
1. Find the gradient of the radius (the line connecting the centre to the point on the circle).
2. Find the perpendicular gradient (the "flip and change sign" trick). This is the gradient of your tangent.
3. Use \(y - y_1 = m(x - x_1)\) with the point on the circle and your new gradient.

6. Intersections: Where Lines and Curves Meet

What happens if a line crosses a curve? To find the intersection points, we use simultaneous equations.

1. Substitute the linear equation (like \(y = mx + c\)) into the curve equation.
2. Rearrange it to get a quadratic equation in the form \(ax^2 + bx + c = 0\).
3. Solve the quadratic equation to find the \(x\)-values, then find the corresponding \(y\)-values.

Interpreting the Discriminant (\(b^2 - 4ac\)):

The discriminant of your resulting quadratic tells you how many times the line and curve meet:
\(b^2 - 4ac > 0\): Two distinct points (the line crosses the curve).
\(b^2 - 4ac = 0\): One point (the line is a tangent to the curve).
\(b^2 - 4ac < 0\): No real roots (the line never touches the curve).

Key Takeaway: If a question mentions a line is a "tangent" to a curve, your first thought should be: "Set the discriminant to zero!"

Quick Review Summary

Straight Lines: Use \(y - y_1 = m(x - x_1)\).
Parallel: \(m_1 = m_2\).
Perpendicular: \(m_1 \times m_2 = -1\).
Circles: \((x - a)^2 + (y - b)^2 = r^2\). Use completion of the square to find the centre.
Tangents: Use the fact that they are perpendicular to the radius.

Don't worry if this seems tricky at first! Coordinate geometry is all about practice. Once you've solved a few problems, you'll start to recognize these "recipes" everywhere!