A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 8: Parametric equations · Lesson 1

Parametric equations and Cartesian form

119 marks

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Date

Do Now

Quick questions to start. Some check what you need today; some are slips that often cost marks.

A1
Needed today
Make $t$ the subject of $x = 3 - 2 ln t$ .
[2 marks]
Answer
A2
Needed today
Write $cos 2 θ$ in terms of $sin θ$ only, and in terms of $cos θ$ only.
[2 marks]
Answer
A3
Common slip
$y = 3 - t^{2}$ for $- 1 < t \leq 2$ . Find the range of values of $y$ .
[2 marks]
Answer
A4
Common slip
Jo writes $tan^{2} θ = sec^{2} θ + 1$ . Correct this identity, then find $tan^{2} θ$ when $sec θ = 3$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

A curve has parametric equations $x = 2 t - 1$ , $y = t^{2} + 3$ . Find a Cartesian equation of the curve.

Hint Make t the subject of the simpler equation, then substitute into the other.

B1
A curve has parametric equations $x = 3 t + 2$ , $y = 2 t^{2} - 1$ , $- 1 \leq t \leq 2$ . Find a Cartesian equation of the curve, and state its domain and range.
[5 marks]
Answer

Example 2

A curve has parametric equations $x = ln (t + 3)$ , $y = \frac{1}{t + 5}$ , $t > - 2$ . Show that the curve is $y = \frac{1}{e ^{x} + 2}$ and state its domain and range.

Hint Undo the ln with e. Then use t > −2 to find the values x and y can take.

B2
A curve has parametric equations $x = ln (t - 1)$ , $y = \frac{4}{t + 1}$ , $t > 2$ . Find a Cartesian equation in the form $y = f (x)$ , and state the domain and range of $f$ .
[5 marks]
Answer

Example 3

A curve has parametric equations $x = 3 + 2 cos t$ , $y = - 1 + 2 sin t$ , $0 \leq t < 2 π$ . Show that the curve is a circle and give its centre and radius.

Hint Make cos t and sin t the subjects, then use sin² t + cos² t = 1.

B3
A curve has parametric equations $x = 5 sin t - 2$ , $y = 5 cos t + 4$ , $0 \leq t \leq π$ . Find a Cartesian equation of the curve and the set of values of $x$ on it. Describe the curve.
[5 marks]

Example 4

A curve has parametric equations $x = sin t$ , $y = cos 2 t$ . Find a Cartesian equation and the domain.

Hint Choose the form of cos 2t that uses only sin t.

B4
A curve has parametric equations $x = 2 cos t$ , $y = 3 cos 2 t + 1$ . Find a Cartesian equation in the form $y = f (x)$ , and state the domain of $f$ .
[4 marks]
Answer

Example 5

A curve has parametric equations $x = sin t$ , $y = sin 2 t$ , $0 \leq t \leq \frac{π}{2}$ . Find a Cartesian equation.

Hint sin 2t = 2 sin t cos t. Find cos t from sin t, and use the interval to choose the sign.

B5
A curve has parametric equations $x = 2 tan t$ , $y = 3 sec t$ , $0 \leq t < \frac{π}{2}$ . Find a Cartesian equation in the form $y = f (x)$ , and state the domain of $f$ .
[5 marks]
Answer

Practice

A-Level

C1
A curve has parametric equations $x = t^{2} - 1$ , $y = 4 t$ , for all real $t$ . Max writes " $y = 4 x + 1$ ". Explain what is wrong, and give a correct Cartesian equation and domain.
[4 marks]

C2
A curve has parametric equations $x = \frac{2}{t - 1}$ , $y = \frac{t + 3}{t - 1}$ , $t > 1$ . Show that the curve is part of a straight line, and find the domain and range.
[5 marks]
Answer
C3
A curve has parametric equations $x = cos t + 2$ , $y = cos 2 t$ . Find a Cartesian equation in the form $y = a x^{2} + b x + c$ , and state the domain and range.
[5 marks]
Answer
C4
A curve has parametric equations $x = 2 sec t$ , $y = tan t + 1$ . Ana writes " $\frac{x ^{2}}{4} = (y - 1)^{2} - 1$ ". Find her mistake and give the correct Cartesian equation in the form $x^{2} = f (y)$ .
[4 marks]
C5
A curve has parametric equations $x = 3 cot t$ , $y = 2 cosec t$ , $0 < t < π$ . Find a Cartesian equation in the form $y = f (x)$ .
[5 marks]
Answer
C6
A curve has parametric equations $x = tan t$ , $y = cos 2 t$ , $- \frac{π}{2} < t < \frac{π}{2}$ . Show that $y = \frac{1 - x ^{2}}{1 + x ^{2}}$ .
[4 marks]

Exam-style questions

Edexcel exam style

D1
A curve C has parametric equations $x = 3 ln (t + 2)$ , $y = t^{2} - 1$ , $t > - 1$ .
(a)
Show that a Cartesian equation of C is $y = e^{\frac{2 x}{3}} - 4 e^{\frac{x}{3}} + 3$ .
[3 marks]
(b)
State the domain of C.
[2 marks]
Answer
(c)
Find the range of C.
[2 marks]
Answer
(d)
Find the exact $x$ -coordinate of the point where C meets the $x$ -axis.
[2 marks]
Answer
Total for D1: 9 marks

D2
Sam writes: " $x = 1 + sin t$ , so $sin t = x + 1$ and $y = 1 - 2 (x + 1)^{2}$ ."
Sam is wrong.
(a)
Find the mistake, and give a correct Cartesian equation of C.
[3 marks]
(b)
Find the domain and range of C.
[3 marks]
Answer
(c)
Find the coordinates of the points where C meets the line $y = x - 1$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
A circle has parametric equations $x = 2 + 5 cos t$ , $y = - 1 + 5 sin t$ , $0 \leq t < 2 π$ . The line $y = x + 2$ meets the circle at two points. By substituting the parametric equations into $y = x + 2$ , find the values of $t$ at these points and the coordinates of the points.
[7 marks]
Answer
D4
The figure shows the curve C with parametric equations $x = 2 sin t$ , $y = sin 2 t$ , $0 \leq t < 2 π$ .
(a)
Show that a Cartesian equation of C is $4 y^{2} = x^{2} (4 - x^{2})$ .
[4 marks]
(b)
Find the coordinates of the points where C meets the coordinate axes.
[3 marks]
Answer
(c)
Find the greatest value of $y$ on C and the value of $x$ where it happens.
[2 marks]
Answer
Total for D4: 9 marks
D5
A curve C has parametric equations $x = 2 - \frac{1}{t}$ , $y = \frac{1}{t ^{2}} + 3$ , $t > 0$ .
(a)
Show that C is part of the curve $y = x^{2} - 4 x + 7$ .
[3 marks]
(b)
State the domain of C.
[2 marks]
Answer
(c)
Explain why each horizontal line $y = k$ meets C at most once.
[2 marks]
(d)
The line $y = 2 x + 3$ meets C at P. Find the exact value of $t$ at P.
[3 marks]
Answer
Total for D5: 10 marks
D6
An ellipse has parametric equations $x = 3 cos t$ , $y = 2 sin t$ , $0 \leq t < 2 π$ .
(a)
Find a Cartesian equation of the ellipse.
[2 marks]
Answer
(b)
Find the exact coordinates of the point where $t = \frac{π}{3}$ .
[2 marks]
Answer
(c)
The line $2 x + 3 y = 6$ meets the ellipse at two points. Find the values of $t$ at these points and their coordinates.
[4 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A curve C has parametric equations $x = sin t + cos t$ , $y = sin 2 t$ , $0 \leq t \leq π$ . Find a Cartesian equation of C and its domain. Hence find the exact coordinates of every point where C meets the line $y = x$ .
[8 marks]
Hint 1 · What to try
Square $x$ . Compare $(sin t + cos t)^{2}$ with $sin 2 t$ .
Hint 2 · The first line
For the domain, write $sin t + cos t$ as $2 sin (t + \frac{π}{4})$ and look at $t + \frac{π}{4}$ for $0 \leq t \leq π$ .
Hint 3 · The full method
Solve $x^{2} - 1 = x$ , then check each root against the domain.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can eliminate the parameter by making t the subject and substituting.
I can eliminate the parameter with a trig identity such as cos 2t = 1 − 2 sin² t.
I can find the domain and range of a parametric curve from the values of the parameter.
I can say which part of a Cartesian curve a parametric curve gives.

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