The coordinate geometry of curves

Welcome to the Coordinate Geometry of Curves!

In your previous studies, you’ve mastered the art of the straight line. But the real world isn’t just made of straight edges—it’s full of curves, orbits, and arches! In this chapter, we are going to explore the beautiful mathematics of circles and other curves. We’ll learn how to describe them using algebra and how to find exactly where they meet. Don’t worry if this seems tricky at first; we’ll take it one step at a time!

1. The Algebra of a Circle

The most famous curve in coordinate geometry is the circle. While a straight line is defined by its gradient, a circle is defined by two things: its centre and its radius.

The Standard Equation

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

Think of it this way: This formula is actually just the Pythagorean theorem in disguise! It says that the distance from any point \((x, y)\) on the edge to the centre \((a, b)\) is always equal to \(r\).

Finding the Centre and Radius

Sometimes, the equation isn't given in the neat format above. It might look like a long string of terms: \(x^2 + y^2 - 4x + 6y - 12 = 0\). To find the centre and radius, we use a technique called completing the square for both \(x\) and \(y\).

Step-by-Step: Completing the Square for Circles
1. Group the \(x\) terms together and the \(y\) terms together.
2. Move any plain numbers (constants) to the other side of the equals sign.
3. Complete the square for the \(x\) part and the \(y\) part separately.
4. Simplify the numbers on the right side to find \(r^2\).

Common Mistake to Avoid: When looking at \((x - 3)^2 + (y + 2)^2 = 25\), students often think the centre is \((-3, 2)\). Remember, the signs are flipped! The centre is actually \((3, -2)\), and the radius is \(\sqrt{25} = 5\).

Quick Review:
• Standard form: \((x - a)^2 + (y - b)^2 = r^2\)
• Centre: \((a, b)\)
• Radius: \(\sqrt{\text{the number on the right}}\)

2. Geometric Properties of Circles

The MEI syllabus expects you to use three specific geometric "shortcuts" to solve coordinate problems. These are very powerful when you need to find equations of tangents or chords.

Property 1: The Tangent and the Radius
The radius of a circle at a given point is perpendicular (\(90^\circ\)) to the tangent at that point.
Memory Trick: Think of a "T" shape where the tangent meets the radius.

Property 2: The Semicircle Property
The angle in a semicircle is always a right angle. If you draw a triangle using the diameter as one side and any point on the edge as the third vertex, it’s always a \(90^\circ\) triangle.

Property 3: The Chord Bisector
The perpendicular line from the centre of a circle to a chord always bisects (cuts in half) that chord.

Did you know?
Navigators and architects have used these properties for thousands of years to ensure accuracy in building domes and plotting courses across the sea!

Key Takeaway: Use these properties to find gradients. If you know the gradient of the radius (\(m_1\)), the gradient of the tangent is its negative reciprocal: \(m_2 = -\frac{1}{m_1}\).

3. Points of Intersection

Often, you’ll be asked to find where a line meets a curve or where two curves cross paths. In coordinate geometry, "where they meet" is just another way of saying "solve simultaneous equations."

Line and Curve

To find where a line (like \(y = x + 2\)) meets a circle, substitute the line equation into the circle equation. This will usually give you a quadratic equation to solve.

How many points?
• If the quadratic has two real solutions (Discriminant \(b^2 - 4ac > 0\)), the line crosses the curve twice.
• If it has one real solution (\(b^2 - 4ac = 0\)), the line is a tangent to the curve.
• If it has no real solutions (\(b^2 - 4ac < 0\)), the line and curve never touch.

4. Parametric Equations

Usually, we describe curves using \(x\) and \(y\) (Cartesian form). But sometimes, it's easier to introduce a third variable, called a parameter, usually written as \(t\) or \(\theta\).

Analogy: Imagine a person walking along a curved path. Their \(x\)-position (how far east they are) depends on time (\(t\)), and their \(y\)-position (how far north they are) also depends on time (\(t\)).

Parametric Form of a Circle

A circle with centre \((a, b)\) and radius \(r\) can be written as:
\(x = a + r\cos(t)\)
\(y = b + r\sin(t)\)

Converting back to Cartesian Form

To go from parametric to Cartesian, you need to "eliminate" the parameter.
• For linear equations: Rearrange one for \(t\) and substitute it into the other.
• For circles/trig: Use the identity \(\sin^2(t) + \cos^2(t) = 1\).

Summary: Parametric equations give \(x\) and \(y\) separately in terms of a third variable, \(t\).

5. Calculus with Parametric Equations

What if you need to find the gradient (\(\frac{dy}{dx}\)) of a curve, but it's written in parametric form? You don't have to convert it back to \(x\) and \(y\) first!

You can use the Chain Rule formula:
\( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \)

Step-by-Step Gradient Finding:
1. Differentiate your \(y\) equation with respect to \(t\) to get \(\frac{dy}{dt}\).
2. Differentiate your \(x\) equation with respect to \(t\) to get \(\frac{dx}{dt}\).
3. Divide the first result by the second result.
4. If you need the gradient at a specific point, plug in the value of \(t\).

Quick Review Box:
To find \(\frac{dy}{dx}\), think "y on top, x on bottom": \( \frac{dy/dt}{dx/dt} \).

6. Real-World Modelling

Parametric equations are incredibly useful in Mechanics, specifically for projectiles. If you kick a ball, its horizontal position (\(x\)) and vertical position (\(y\)) both change over time (\(t\)).

By eliminating \(t\), you can find the "Equation of the Trajectory"—which is the Cartesian path the ball follows through the air. This path is almost always a parabola (a quadratic curve).

Key Takeaway for Modelling:
• \(t\) usually represents time.
• The domain of \(t\) might be restricted (e.g., \(t \ge 0\) because you can't have negative time).
• Intersection points in models often represent where an object hits a target or the ground.

Final Encouragement: Coordinate geometry is all about turning pictures into algebra. If you get stuck, always draw a sketch. Seeing the curve makes the equations much easier to handle!