Parametric Equations

Expressing curves using a parameter to define x and y coordinates separately

SPEC 3.3 • COORDINATE GEOMETRY

What Are Parametric Equations?

Definition

Parametric equations express the x and y coordinates of points on a curve as separate functions of a third variable called the parameter, usually denoted by t.

x = f (t), y = g (t)

As t varies, the point (x, y) traces out a curve in the xy-plane.

Key Concept

The parameter t is often interpreted as time, so parametric equations naturally describe motion along a path. However, t can represent any quantity.

Why Use Parametric Form?

Advantage 1

Describes motion: Shows HOW a curve is traced, including direction and speed

Advantage 2

Handles complex curves: Some curves cannot be written as y = f(x)

Advantage 3

Multi-valued functions: A circle can't be written as a single function y = f(x)

Advantage 4

Easier differentiation: Finding gradients via

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

Example

A circle x² + y² = 25 cannot be written as y = f(x) for all points, but parametrically: x = 5cos t, y = 5sin t traces the entire circle as t goes from 0 to 2π.

Standard Parametric Curves

CurveParametric FormCartesian Form

Circle (centre O, radius r)

x = r cos t, y = r sin t

x^{2} + y^{2} = r^{2}

Circle (centre (a,b), radius r)

x = a + r cos t, y = b + r sin t

(x - a)^{2} + (y - b)^{2} = r^{2}

Ellipse

x = a cos t, y = b sin t

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

Parabola

x = a t^{2}, y = 2 a t

y^{2} = 4 a x

Important

For trigonometric parametric equations, use the identity

sin^{2} t + cos^{2} t = 1

to eliminate the parameter.

Converting to Cartesian Form

To convert parametric equations to Cartesian form, eliminate the parameter t.

Method 1: Substitution

Make t the subject from one equation, substitute into the other.

Method 2: Trigonometric Identity

For

x = r cos t, y = r sin t

: use

cos^{2} t + sin^{2} t = 1

(\frac{x}{r})^{2} + (\frac{y}{r})^{2} = 1 ⟹ x^{2} + y^{2} = r^{2}

Method 3: Algebraic Manipulation

Square, add, or manipulate equations to eliminate t directly.

Worked Example01

Convert the parametric equations $x = 2 t, y = t^{2} - 1$ to Cartesian form.

1

From x = 2t, make t the subject:

t = \frac{x}{2}

2

Substitute into y = t² − 1:

y = (\frac{x}{2})^{2} - 1

3

Simplify:

y = \frac{x ^{2}}{4} - 1

Answer

y = \frac{x ^{2}}{4} - 1

(a parabola)

Worked Example02

Convert $x = 3 cos t, y = 3 sin t$ to Cartesian form and identify the curve.

1

From x = 3cos t:

cos t = \frac{x}{3}

2

From y = 3sin t:

sin t = \frac{y}{3}

3

Use identity

sin^{2} t + cos^{2} t = 1

:

(\frac{x}{3})^{2} + (\frac{y}{3})^{2} = 1

4

Simplify:

\frac{x ^{2}}{9} + \frac{y ^{2}}{9} = 1 ⟹ x^{2} + y^{2} = 9

Answer

x^{2} + y^{2} = 9

— a circle with centre (0, 0) and radius 3

Worked Example03

Determine the Cartesian equation of the curve with parametric equations $x = 2 cos t, y = 5 sin t$ . Identify the curve.

1

From x = 2cos t:

cos t = \frac{x}{2}

2

From y = 5sin t:

sin t = \frac{y}{5}

3

Use identity:

(\frac{x}{2})^{2} + (\frac{y}{5})^{2} = 1

4

Cartesian equation:

\frac{x ^{2}}{4} + \frac{y ^{2}}{25} = 1

Answer

\frac{x ^{2}}{4} + \frac{y ^{2}}{25} = 1

— an ellipse with semi-axes a = 2 (horizontal) and b = 5 (vertical)

Sketching Parametric Curves

Method

Create a table of values for t, x, and y

Plot each point (x, y) on the coordinate axes

Connect points in order of increasing t

Add arrows to show direction of travel

Note any key features (start, end, turning points)

Important

The direction of the curve matters! Always indicate with arrows which way the curve is traced as t increases.

Worked Example04

Sketch the curve $x = t^{2}, y = t^{3}$ for $- 2 \leq t \leq 2$ . Include direction arrows.

1

Create table of values:

2

Points: (4, −8), (1, −1), (0, 0), (1, 1), (4, 8)

3

Plot and connect in order. Curve passes through origin, symmetric about y-axis in x but not y.

AnswerCurve goes from (4, −8) through origin to (4, 8). Arrow points upward as t increases. This is a semicubical parabola:

y^{2} = x^{3}

Domain Restrictions

The range of the parameter t determines which portion of the curve is traced.

Key Principle

Restricting t restricts the curve. Different t-ranges give different portions of the same Cartesian curve.

Example: Circle

x = cos t, y = sin t

0 ≤ t ≤ 2π: Full circle

0 ≤ t ≤ π: Upper semicircle

0 ≤ t ≤ π/2: Quarter circle (Q1)

Finding t for a Point

Given a point on the curve, substitute into parametric equations and solve for t.

Worked Example05

A curve has parametric equations $x = 4 cos t, y = 4 sin t$ for $0 \leq t \leq \frac{π}{2}$ . Describe the portion of the curve traced.

1

Cartesian equation: x² + y² = 16 (circle, centre O, radius 4)

2

At t = 0: x = 4cos(0) = 4, y = 4sin(0) = 0 → Point (4, 0)

3

At t = π/2: x = 4cos(π/2) = 0, y = 4sin(π/2) = 4 → Point (0, 4)

4

The curve traces from (4, 0) to (0, 4) anticlockwise.

AnswerQuarter circle in the first quadrant, from (4, 0) to (0, 4), traced anticlockwise.

Points of Intersection

To find where a parametric curve intersects a line or another curve:

Method

Substitute x = f(t) and y = g(t) into the equation of the line/curve

Solve the resulting equation for t

Find x and y values for each value of t

State the coordinates of intersection points

Check

If t is restricted to a certain range, only include values of t within that range.

Worked Example06

Find the coordinates where the curve $x = t, y = t^{2}$ meets the line $y = 2 x + 3$ .

1

Substitute parametric equations into y = 2x + 3:

2

t^{2} = 2 t + 3

3

Rearrange:

t^{2} - 2 t - 3 = 0

4

Factorise:

(t - 3) (t + 1) = 0

, so t = 3 or t = −1

5

When t = 3: x = 3, y = 9 → Point (3, 9)

6

When t = −1: x = −1, y = 1 → Point (−1, 1)

AnswerThe curve intersects the line at (3, 9) and (−1, 1)

Practice

Question 1: Convert the parametric equations

x = t + 1, y = t^{2} - 2 t

to Cartesian form.

1

From x = t + 1: t = x − 1

2

Substitute: y = (x − 1)² − 2(x − 1)

3

Expand: y = x² − 2x + 1 − 2x + 2 = x² − 4x + 3

Answer

y = x^{2} - 4 x + 3

Practice

Question 2: A curve has parametric equations

x = 5 cos t, y = 3 sin t

. State the Cartesian equation and identify the curve.

1

cos t = x/5 and sin t = y/3

2

Use identity: (x/5)² + (y/3)² = 1

3

Cartesian equation:

\frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1

Answer

\frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1

— an ellipse with semi-major axis 5 (horizontal) and semi-minor axis 3 (vertical)

Practice

Question 3: Find the coordinates where the curve

x = 2 t, y = \frac{8}{t}

intersects the line

y = x

.

1

Substitute into y = x:

\frac{8}{t} = 2 t

2

Multiply by t: 8 = 2t²

3

t² = 4, so t = ±2

4

When t = 2: x = 4, y = 4 → (4, 4)

5

When t = −2: x = −4, y = −4 → (−4, −4)

Answer(4, 4) and (−4, −4)

Summary

Key Concepts

TopicKey Points

Parametric Formx = f(t), y = g(t) where t is the parameter

Converting to CartesianEliminate t using substitution or identities

Trig CurvesUse sin²t + cos²t = 1 for circles/ellipses

Domain Restrictionst-range determines portion of curve traced

IntersectionsSubstitute parametric form into other equation, solve for t

Exam Skills

Convert parametric equations to Cartesian form fluently

Recognise standard parametric forms (circle, ellipse, parabola)

Determine the portion of curve from parameter restrictions

Find intersection points by solving for the parameter