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

Sketching curves and points of intersection

113 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
Find the coordinates of the points where the line $y = 2 x + 1$ meets the circle $x^{2} + y^{2} = 13$ .
[2 marks]
Answer
A2
Needed today
Solve $sin t + cos t = 1$ for $0 \leq t < 2 π$ .
[2 marks]
Answer
A3
Last lesson
A curve has parametric equations $x = t^{2} + 1$ , $y = 2 t$ . Find the value of $t$ at the point $(5, - 4)$ .
[2 marks]
Answer
A4
Common slip
A curve has parametric equations $x = cos t$ , $y = sin t$ , $0 \leq t \leq π$ . Describe the curve.
[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

Sketch the curve with parametric equations $x = t - 1$ , $y = t^{2} - 4$ , $- 1 \leq t \leq 3$ . Mark the end points and where it meets the axes.

Hint Find the points at the two ends of the interval. Then set x = 0 and y = 0, and keep only values of t in the interval.

B1
Sketch the curve with parametric equations $x = t + 2$ , $y = t^{2} - 1$ , $- 2 \leq t \leq 2$ . Mark the end points, the lowest point and where it meets the axes.
[5 marks]

Example 2

A curve has parametric equations $x = t^{2} - 4$ , $y = t^{3} - t$ . Find the points where it meets the coordinate axes.

Hint Set y = 0 and solve for every t, then find x. Do the same with x = 0.

B2
A curve has parametric equations $x = t^{2} - 1$ , $y = t^{3} - 4 t$ . Find the coordinates of the points where it meets the coordinate axes.
[4 marks]
Answer

Example 3

A curve has parametric equations $x = 2 t^{2} - 1$ , $y = 3 t$ . Find where it meets the line $x - y = 4$ .

Hint Put the parametric x and y into the equation of the line. Solve for t, then find the points.

B3
A curve has parametric equations $x = t^{2}$ , $y = 2 t$ . Find the coordinates of the points where it meets the line $y = x - 3$ .
[4 marks]
Answer

Example 4

A curve has parametric equations $x = 2 sin t$ , $y = 3 cos 2 t$ , $0 \leq t < 2 π$ . Find the points where it meets the line $y = \frac{3}{2}$ .

Hint Solve cos 2t = 1/2 for 2t in the interval 0 ≤ 2t < 4π, then halve.

B4
A curve has parametric equations $x = 3 cos t$ , $y = cos 2 t + 1$ , $0 \leq t < 2 π$ . Find the values of $t$ , and the points, where it meets the line $y = \frac{1}{2}$ .
[5 marks]
Answer

Example 5

A curve has parametric equations $x = t + k$ , $y = k t^{2}$ , where $k$ is a constant. The curve passes through $(3, 4)$ . Find the possible values of $k$ .

Hint Make t the subject of the x equation at the point, then put it into the y equation.

B5
A curve has parametric equations $x = t - k$ , $y = k t$ , where $k$ is a constant. The curve passes through $(1, 6)$ . Find the possible values of $k$ , and the value of $t$ at the point for each.
[4 marks]
Answer

Practice

A-Level

C1
A curve has parametric equations $x = 3 t$ , $y = t^{2}$ . Find the coordinates of the points where it meets the line $y = x - 2$ .
[4 marks]
Answer

C2
A curve has parametric equations $x = 2 cos t$ , $y = 2 sin t$ , $\frac{π}{2} \leq t \leq \frac{3 π}{2}$ . Lee says the curve is a circle. Describe the curve correctly and give its end points.
[4 marks]
C3
A curve has parametric equations $x = t^{2}$ , $y = t^{3} - 4 t$ . Kai divides $t^{3} - 4 t = 0$ by $t$ , gets $t = \pm 2$ , and says the curve meets the $x$ -axis only at $(4, 0)$ . Find every point where the curve meets the coordinate axes.
[4 marks]
Answer
C4
A curve has parametric equations $x = t^{2} - 2 t$ , $y = t^{2} + t - 1$ . Show that $(3, - 1)$ lies on the curve, and decide whether $(3, 5)$ lies on the curve.
[5 marks]
C5
A curve has parametric equations $x = t^{2}$ , $y = t^{3} - 3 t$ . The curve crosses itself at one point. Find the two values of $t$ there and the coordinates of the point.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A curve C has parametric equations $x = ln (t + 1)$ , $y = t^{2} - 4$ , $t > - 1$ .
(a)
Show that a Cartesian equation of C is $y = e^{2 x} - 2 e^{x} - 3$ .
[3 marks]
(b)
Find the exact coordinates of the points where C meets the coordinate axes.
[4 marks]
Answer
(c)
Find the equation of the asymptote to C.
[2 marks]
Answer
Total for D1: 9 marks

D2
A curve C has parametric equations $x = 3 t^{2}$ , $y = 6 t$ .
(a)
Find a Cartesian equation of C.
[2 marks]
Answer
(b)
Show that the line $y = x + 3$ meets C at exactly one point, and find its coordinates.
[3 marks]
(c)
The line $y = 2 x - 12$ meets C at A and B. Find the coordinates of A and B.
[3 marks]
Answer
(d)
Find the exact length AB.
[2 marks]
Answer
Total for D2: 10 marks
D3
Mia says: " $sin t = \frac{1}{2}$ gives $t = \frac{π}{6}$ , so C meets the line $y = \frac{1}{2}$ at one point only."
Mia is wrong.
(a)
Find the mistake, and find the coordinates of every point where C meets the line $y = \frac{1}{2}$ .
[3 marks]
(b)
Find a Cartesian equation of C, and describe C.
[2 marks]
Answer
(c)
The line $x + y = 3$ meets C at two points. Find the values of $t$ at these points and the coordinates of the points.
[4 marks]
Answer
Total for D3: 9 marks
D4
A circle has parametric equations $x = 3 + 5 cos t$ , $y = - 2 + 5 sin t$ , $0 \leq t < 2 π$ . The line $y = 2 x + k$ , where $k$ is a constant, is a tangent to the circle. Find the possible values of $k$ , giving your answers in the form $a + b 5$ .
[7 marks]
Answer
D5
The figure shows the curve C with parametric equations $x = cos t$ , $y = sin 2 t$ , $0 \leq t < 2 π$ .
(a)
Show that C passes through the origin twice, and give the two values of $t$ there.
[2 marks]
(b)
Find the coordinates of every point where C meets the line $y = x$ .
[4 marks]
Answer
(c)
Find the exact coordinates of the points where C meets the line $y = 1$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
A curve C has parametric equations $x = t^{3}$ , $y = t^{2}$ .
(a)
Find the coordinates of the points where C meets the line $y = 2 x$ .
[3 marks]
Answer
(b)
The line $y = k x$ , where $k > 0$ , meets C at the origin and at one other point P. Find the coordinates of P in terms of $k$ .
[2 marks]
Answer
(c)
P lies on the line $x + y = 2$ . Find the value of $k$ .
[4 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

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

E1
The figure shows the curve $C_{1}$ with parametric equations $x = 3 cos t$ , $y = 3 sin t$ , and the curve $C_{2}$ with parametric equations $x = 4 cos s$ , $y = 2 sin s$ , where $0 \leq t < 2 π$ and $0 \leq s < 2 π$ . The two curves meet at four points, which are the corners of a rectangle. Find the exact area of the rectangle, giving your answer in the form $\frac{a 35}{b}$ .
[8 marks]
Hint 1 · What to try
Write a Cartesian equation for each curve.
Hint 2 · The first line
Both equations hold at a corner. Subtract to remove $x^{2}$ and find $y^{2}$ , then $x^{2}$ .
Hint 3 · The full method
By symmetry the corners are $(\pm x, \pm y)$ , so the sides are $2 x$ and $2 y$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch a parametric curve, marking its end points and where it meets the axes.
I can find where a parametric curve meets the axes by setting x = 0 or y = 0.
I can find where a parametric curve meets a line by substituting x and y into its equation.
I can check every value of the parameter against the interval given.

Answers and mark scheme

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