Equations

a x + b y = c

p x + q y = r

Parametric Equations

x = r cos θ

y = r sin θ

Parametric Equations

Parametric equations are a set of explicit equations that represent a mapping that is not a function, i.e. a many-to-many mapping or a one-to-many mapping. These can be useful in solving problems using the methods known for handing functions.

A curve, $C$ can be defined using a set of parametric equations $x = u (θ)$ and $y = v (θ)$ . Each value of $θ$ defines a point on $C$ with coordinates $(u (θ), v (θ))$ .

For a given set of parametric equations a cartesian equation can be formed by using substitution to eliminate the parameter.

For parametric equations $x = u (θ)$ and $y = v (θ)$ with Cartesian equation $y = f (x)$ :

the domain of $y = f (x)$ is the range of $x = u (θ)$ ;
the range of $y = f (x)$ is the range of $y = v (θ)$ .

Questions:

A level Edexcel Year 2 Text book chapter 8 page 197

Solution Bank

Parametric equations
(Care of alevelmaths.co.uk)

Parametric equations
(Care of www.physicsandmathstutor.co.uk)

Parametric Curves
(Care of www.physicsandmathstutor.co.uk)

Exam parametric equations
(Care of www.examsolutions.net)

(1)

Figure 1.1

Figure 1.1 shows a sketch of the curve $C$ with parametric equations

x = 4 cos (t + \frac{π}{6}),

y = 2 sin t,

0 < t \leq 2 π

Show that the Cartesian equation of $C$ can be written in the form:

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

Where $a$ and $b$ are constants to be found.

[ Solution 1 ]

x + y = 4 cos (t + \frac{π}{6}) + 2 sin t

x + y = 4 (cos t cos \frac{π}{6} - sin t sin \frac{π}{6}) + 2 sin t

x + y = 4 (\frac{\sqrt{3}}{2} cos t - \frac{1}{2} sin t) + 2 sin t

x + y = 2 \sqrt{3} cos t

{(x + y)}^{2} = 12 {cos}^{2} t

{(x + y)}^{2} = 12 (1 - {sin}^{2} t)

{(x + y)}^{2} = 12 - 3 y^{2}

{(x + y)}^{2} + 3 y^{2} = 12

a = 3, b = 12

Parametric Equations
(Care of alevelmaths.co.uk)

Exam Questions - Parametric Equations
(Care of www.examsolutions.net)

Next Steps:

Parametric Differentiation
(Care of revisionmaths.com)

Parametric Integration

Simultaneous Equations

Warm-up

Rearranging Formulae
(Care of mmerevise.co.uk)

Graphical
Textbook Exercise

Worked Examples:

Example 1:

Solve these two simultaneously using elimination:

y = 3 x + 7

4 x + y = 14

Aligning the terms of each equation first

y = 3 x + 7

4 x + y = 14

subtracting

-4 x = 3 x - 7

Simplifying

7 = 7 x

x = 1

Substituting this into the first equation in the question,

y = 3 \times 1 + 7 = 10

Checking by substituting both into the second equation.

4 \times 1 + 10 = 14

Example 2:

Solve these two simultaneously:

3 x + 5 y = 13

5 x - y = 3

Multiplying the second equation by five,

25 x - 5 y = 15

Then adding the third to the first equation,

28 x = 28

So $x = 1$

Substituting this result into the second equation gives $y = 2$

Example 3:

Solve by substitution:

5 x + 3 y = 41

2 x + 3 y = 20

Making $3 y$ the subject of the first equation:

3 y = 41 - 5 x

The substituting this into the second:

2 x + 41 - 5 x = 20

-3 x = -21

x = 7

Substituting $x = 7$ into the first equation:

5 \times 7 + 3 y = 41

3 y = 6

y = 2

Example 4:

Solve these two simultaneously:

x^{2} + y^{2} = 20

x + y = 6

Rearrange the second equation to give $y$ as the subject

y = 6 - x

and substituting in the first equation and solving

x^{2} + {(6 - x)}^{2} = 20

x^{2} + 36 - 12 x + x^{2} = 20

x^{2} - 6 x + 8 = 0

x = 2

y = 4

x = 4

y = 2

Questions and Further Materials

Simultaneous Equations
(care of www.physicsandmathstutor.com)

Simultaneous equations
(care of www.mathsgenie.co.uk)

Elimination and Substitution, both linear: Practice Questions; Textbook Exercise
Advanced inc quadratic: Practice Questions 1; Practice Questions 2 [ MS ].; Textbook Exercise
3 Unknowns: Textbook Exercise; (care of corbettmaths.com); Solving simultaneous equations; (care of www.mrbartonmaths.com)

Linear Equations
$y = m x + c$

Functions
$f (x)$

Quadratic Equations
$y = a x^{2} + b x + c$

Simultaneous Equations

Parametric Equations

Parametric Equations

Questions:

A level Edexcel Year 2 Text book chapter 8 page 197

Solution Bank

Parametric equations
(Care of alevelmaths.co.uk)

Parametric equations
(Care of www.physicsandmathstutor.co.uk)

Parametric Curves
(Care of www.physicsandmathstutor.co.uk)

Exam parametric equations
(Care of www.examsolutions.net)

(1)

[ Solution 1 ]

Further Reading:

Parametric Equations
(Care of alevelmaths.co.uk)

Exam Questions - Parametric Equations
(Care of www.examsolutions.net)

Next Steps:

Parametric Differentiation
(Care of revisionmaths.com)

Parametric Integration

Simultaneous Equations

Warm-up

Rearranging Formulae
(Care of mmerevise.co.uk)

Graphical
Textbook Exercise

Worked Examples:

Example 1:

Example 2:

Example 3:

Example 4:

Questions and Further Materials

Parametric Equations

Questions:

A level Edexcel Year 2 Text book chapter 8 page 197

Solution Bank

Parametric equations (Care of alevelmaths.co.uk)

Parametric equations (Care of www.physicsandmathstutor.co.uk)

Parametric Curves (Care of www.physicsandmathstutor.co.uk)

Exam parametric equations (Care of www.examsolutions.net)

(1)

[ Solution 1 ]

Further Reading:

Parametric Equations (Care of alevelmaths.co.uk)

Exam Questions - Parametric Equations (Care of www.examsolutions.net)

Next Steps:

Parametric Differentiation (Care of revisionmaths.com)

Simultaneous Equations

Warm-up

Rearranging Formulae (Care of mmerevise.co.uk)

Worked Examples:

Example 1:

Example 2:

Example 3:

Example 4:

Questions and Further Materials

Parametric equations
(Care of alevelmaths.co.uk)

Parametric equations
(Care of www.physicsandmathstutor.co.uk)

Parametric Curves
(Care of www.physicsandmathstutor.co.uk)

Exam parametric equations
(Care of www.examsolutions.net)

Parametric Equations
(Care of alevelmaths.co.uk)

Exam Questions - Parametric Equations
(Care of www.examsolutions.net)

Parametric Differentiation
(Care of revisionmaths.com)

Rearranging Formulae
(Care of mmerevise.co.uk)