Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WMA14 · Pure Mathematics 4Coordinate geometry in the (x, y) planeParametric equations

Coordinate geometry in the (x, y) plane · Subtopic 1 of 1

Parametric Equations

A curve can be described by giving $x$ and $y$ separately in terms of a parameter $t$ . Eliminate $t$ (by substitution or a trig identity) to get the Cartesian equation, and substitute $x(t)$ , $y(t)$ into a line's equation to find intersections.

Edexcel YMA01 WMA14≈ 35 minRevision 1 · updated 2026-10-27

Practise this topicTimed test Ask Salam Tutor

Section 1 of 14

Learning objectives

Plot and interpret curves given parametrically.
Convert parametric equations to Cartesian form.
Find where a parametric curve meets a line.

Pause and think

A ball's position after $t$ seconds is $(10t,\ 20t-5t^2)$ . What extra information does this give compared with $y=2x-\frac{x^2}{20}$ ?

Section 2 of 14

Core idea

Start hereWhere and when

A ball's path through the air is a curve, but the curve alone doesn't say when the ball is at each point. Give $x$ and $y$ separately as functions of time and you have both. That's a parametric equation.

The big idea

A curve given by $x=f(t)$ , $y=g(t)$ can be converted to Cartesian form by eliminating $t$ : rearrange one equation for $t$ , or use an identity such as $\cos^2t+\sin^2t=1$ .

1. Eliminating the parameter

Worked exampleSubstitution

Find the Cartesian equation of $x=2t$ , $y=t^2$ .

Goal: $y$ in terms of $x$ .
Strategy: Make $t$ the subject of the simpler equation.

$t=\frac x2$
From $x=2t$ .
$y=\left(\frac x2\right)^2$
Substitute.

Answer

y=\frac{x^2}4

✓ Check

t=3

(6,9)

and

\frac{36}4=9

✓.

Circle from

x=a+r\cos t

y=b+r\sin t

$\cos t=\frac{x-a}r,\ \sin t=\frac{y-b}r$
Isolate the trig functions.
$\left(\frac{x-a}r\right)^2+\left(\frac{y-b}r\right)^2=1$
$\cos^2t+\sin^2t=1$ .
$(x-a)^2+(y-b)^2=r^2$
Multiply by $r^2$ .

Centre

(a,b)

, radius

r

\cos^2t+\sin^2t=1,\qquad\sec^2t-\tan^2t=1

$t$: parameter.

2. Meeting a line

Worked exampleIntersection

Find where $x=t^2$ , $y=2t$ meets $y=x-3$ .

Goal: Coordinates.
Strategy: Substitute both coordinates into the line.

$2t=t^2-3$
Substitute $x(t)$ and $y(t)$ .
$(t-3)(t+1)=0$
Rearrange and factorise.
$t=3:(9,6);\ t=-1:(1,-2)$
Convert back to points.

Answer

(9,6)

and

(1,-2)

✓ Check

6=9-3

✓,

-2=1-3

✓.

Choosing a method

Drawing diagram…

Pick the elimination method from how t appears.

Edge caseRestricted range If

0\le t\le\pi

x=\cos t

y=\sin t

, only the upper semicircle is traced: state

y\ge0

with the Cartesian equation.

Common misconceptionAnswer the question Solving for

t

isn't the end: convert to

(x,y)

Pause and think

What curve is $x=\cos t$ , $y=\cos2t$ ?

Summary

Eliminate $t$ by substitution or identity.
Respect the parameter's range.
Intersections: substitute into the other equation.

Parameter: Variable (often $t$ ) both coordinates depend on.
Cartesian equation: Relation between $x$ and $y$ only.

HistoryCycloids. The path of a point on a rolling wheel,

x=r(t-\sin t)

y=r(1-\cos t)

, fascinated Galileo, Pascal and the Bernoullis in the 1600s.

Fun factSpirograph. Spirograph patterns are parametric curves made by circles rolling inside circles.

Section 3 of 14

Key formulae

Circle

x=a+r\cos t,\ y=b+r\sin t\ \Rightarrow\ (x-a)^2+(y-b)^2=r^2

Eliminating t

t=f^{-1}(x)\ \text{or}\ \cos^2t+\sin^2t=1

TrickLook for the identity. If

x

involves

\cos t

and

y

involves

\sin t

, isolate them and use

\cos^2t+\sin^2t=1

. For

\sec

and

\tan

, use

\sec^2t-\tan^2t=1

Section 4 of 14

Worked examples

Example 1 · Linear parameter0 / 2 steps

Find the Cartesian equation of

x=2t

y=t^2

Example 2 · Intersection0 / 3 steps

Where does

x=t^2

y=2t

meet

y=x-3

Section 5 of 14

Explore the graph

A parametric circle

Loading the interactive graph…

Try this

The point moves round the circle centre $(a,b)$ as $s$ increases.
Check $(x-a)^2+(y-b)^2=r^2$ at any position.

$x=a+r\cos t$ , $y=b+r\sin t$ .

In the real worldAnimation. Computer animators move objects along parametric paths, with the parameter as time.

Section 6 of 14

Common mistakes

IncorrectIgnoring the parameter's range

CorrectRestrict the Cartesian curve

t\in[0,\pi]

may give only part of the curve.

IncorrectUsing the identity without isolating

CorrectIsolate

\cos t

and

\sin t

first

x=3\cos t

means

\cos t=\frac x3

IncorrectReporting

t

values as points

CorrectSubstitute back for

(x,y)

The question asks for coordinates.

y = f(x)

Parametric: x and y, each on their own schedule.

Section 7 of 14

Exam tip

Exam tipState the domain. When converting, give the domain of

x

that matches the parameter range.

Section 8 of 14

Guided practice

SA-PM-PAR-G012 marksL1 FoundationNon-calculatorConverting

Find the Cartesian equation of

x=2t

y=t^2

. Give

y

in terms of

x

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-PM-PAR-0011 markL1 FoundationNon-calculatorConverting

x=3\cos t

y=3\sin t

is a circle. Find its radius.

Salam Academy Original Practice · Proposed mark scheme

SA-PM-PAR-0022 marksL1 FoundationNon-calculatorConverting

Find

y

in terms of

x

for

x=t+1

y=2t-3

Salam Academy Original Practice · Proposed mark scheme

SA-PM-PAR-0033 marksL3 ExamNon-calculatorIntersections

The curve

x=t^2

y=2t

meets the line

y=x-3

at two points. Find their

x

-coordinates.

Salam Academy Original Practice · Proposed mark scheme

SA-PM-PAR-0042 marksL2 StandardNon-calculatorConverting

x=1+2\cos t

y=3+2\sin t

. Find the

y

-coordinate of the centre of this circle.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-PM-PAR-0053 marksL3 ExamNon-calculatorConverting

x=2\sec t

y=3\tan t

satisfies

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

. Find

k

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Eliminate $t$ .
Trig identities for trig parameters.
Intersections: substitute into the line.

Section 12 of 14

Quick quiz

Q1 · SA-PM-PAR-Q011 markL1 FoundationNon-calculatorIntersections

x=\cos t

y=\sin t

for

0\le t\le\pi

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics WMA14 (P4)
Where: Coordinate geometry in the (x, y) plane
Expected: Use parametric equations of curves; convert to Cartesian form, including with trig identities; find intersections with lines.
Calculator: Permitted.
Typical style: A typical question converts x = 3 cos t, y = 2 sin t to Cartesian form, then finds intersections.
Open in: YMA01 · WMA14

Section 14 of 14

Past paper questions

No past paper questions are tagged to this subtopic yet. We add papers every week.

Browse past papers

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).