Post on 29-Aug-2018
C4 Maths Parametric equations Page 1
Edexcel past paper questions
Core Mathematics 4
Parametric Equations
Edited by: K V Kumaran
Email: kvkumaran@gmail.com
C4 Maths Parametric equations Page 2
Co-ordinate Geometry A parametric equation of a curve is one which does not give the relationship between x and y directly but rather uses a third variable, typically t, to do so. The third variable is known as the parameter. A simple example of a pair of parametric equations: x = 5t + 3 y = t2 + 2t Converting to Cartesian You need to be able to find the Cartesian equation of the curve from parametric equations, that is the equation that relates x and y directly. To do this you need to eliminate the parameter. The easiest way to do this is to rearrange on parametric equation to get the parameter as the subject and then substitute this into the other equation. A circle with an origin (a, b) has the parametric equation:
𝑥 = 𝑎 + 𝑟𝑐𝑜𝑠𝜃 𝑦 = 𝑏 + 𝑟𝑠𝑖𝑛𝜃
You can use the result sin2 𝜃 + cos2 𝜃 = 1 to derive these. As before, θ is the parameter instead of t in the equations. You need to be able to recognise these as parametric equations of circles in the exam.
C4 Maths Parametric equations Page 3
C4 Maths Parametric equations Page 4
Example The curve C is described by the parametric equations x = 5cost, y = cos2t, 0 ≤ t ≤ π a) Find a Cartesian equation for the curve C. b) Draw a sketch of the curve C. a) From C3 again you should remember that: cos2t cos2t – sin2t 2cos2t – 1 Therefore: y = 2cos2t – 1
If x = 5cost then cost = x
5
So finally y = 22x
25- 1
b) This is simply a quadratic that is symmetrical about the y axis and intercepts with the y axis at -1.
-1
C4 Maths Parametric equations Page 5
Example
C4 Maths Parametric equations Page 6
The parametric equations of a curve are x = 14 sin t, y = 14t cos t
where 0 < t < 2
π. Find
dx
dy in terms of t, and hence show that the gradient of the
curve is zero where t
1ttan
Since the curve is given parametrically we can use the chain rule to find dx
dy
dy dy dt
dx dt dx
So by differentiating the parametric:
x = 14 sin t y = 14t cos t (a Product)
dx
dt= 14 cos t
dy
dt= 14 cos – 14t sin t
dy dy dt
dx dt dx
dy 14 cost - 14t sin t
1 ttantdx 14cost
When the gradient is zero
1 ttant 0
1tant
t
This equation could be solved by iterative methods (C3).
C4 Maths Parametric equations Page 7
Example 4 A curve is given by the parametric equations
x = 7 sin3 t, y = 6 cos 2t, 0 < t < 4
.
Show that tsin8
7
dy
dx
By chain rule
dx dx dt
dy dt dy
2dx21sin tcost
dt
dy12sin2t don't forget the 2.
dt
By C3 trig identities sin 2t = 2 sin t cos t
2dx dx dt 21sin tcost
dy dt dy 24sintcost
dx 7 sint
dy 8
The final example in this section deals with tangents and normals to curves. Example 5
The curve C is described by the parametric equations
x = tan t y = sin 2t 2
t2
a) Find the gradient of the curve at the point P where t= 3
π
b) Find the equation of the normal to the curve at P.
a) Find the gradient of the curve at the point P where t= 3
C4 Maths Parametric equations Page 8
Using chain rule:
tsecdt
dx 2 t2cos2dt
dy
dx dx dt
dy dt dy
t2costcos2tsec
t2cos2
dx
dy 22
Let t= 3
π
Grad = 25.03
2cos
3cos2
2
b) Find the equation of the normal to the curve at P. We are asked for the equation of the normal therefore the gradient will be 4 (why?).
Using t= 3
the x and y coordinates are 3 and
2
3respectively.
Using y = mx + c
2
3 = 4 3 + c
c = 2
37
Therefore the equation of the normal is
2
37x4y
Differentiating ax
C4 Maths Parametric equations Page 9
This function describes growth and decay, and its derivative gives a measure of the rate of change of this growth/decay. Since 𝑦 = 𝑎𝑥, taking logs of both sides gives ln 𝑦 = ln 𝑎𝑥 = 𝑥 ln 𝑎. Using implicit differentiation to differentiate ln 𝑦:
1
𝑦
𝑑𝑦
𝑑𝑥= ln 𝑎
𝑑𝑦
𝑑𝑥= 𝑦 ln 𝑎 = 𝑎𝑥 ln 𝑎
This result needs to be learn, and is not given in the formula sheet.
C4 Parametric differentiation past paper questions
C4 Maths Parametric equations Page 10
1. A curve has parametric equations
x = 2 cot t, y = 2 sin2 t, 0 < t 2
.
(a) Find an expression for x
y
d
d in terms of the parameter t. (4)
(b) Find an equation of the tangent to the curve at the point where t = 4
. (4)
(c) Find a cartesian equation of the curve in the form y = f(x). State the domain on which the curve is defined. (4)
(C4 June 2005, Q6.) 2. Figure 2
The curve shown in Figure 2 has parametric equations
x = sin t, y = sin
6
t ,
2
< t <
2
.
(a) Find an equation of the tangent to the curve at the point where t = 6
.
(6)
(b) Show that a cartesian equation of the curve is
x 1 0.5 –1 –0.5
0.5
y
O
C4 Maths Parametric equations Page 11
y = 2
3x +
2
1(1 – x2), –1 < x < 1.
(3)
(C4 June 2006, Q4.) 3. A curve has parametric equations
x = 7 cos t – cos 7t, y = 7 sin t – sin 7t, 8
< t <
3
.
(a) Find an expression for x
y
d
d in terms of t. You need not simplify your answer.
(3)
(b) Find an equation of the normal to the curve at the point where t = 6
.
Give your answer in its simplest exact form. (6)
(C4 Jan 2007, Q3.) 4. A curve has parametric equations
x = tan2 t, y = sin t, 0 < t < 2
.
(a) Find an expression for x
y
d
d in terms of t. You need not simplify your answer.
(3)
(b) Find an equation of the tangent to the curve at the point where t = 4
.
Give your answer in the form y = ax + b , where a and b are constants to be
determined. (5)
(c) Find a cartesian equation of the curve in the form y2 = f(x). (4)
(C4 June 2007, Q6.) 5.
C4 Maths Parametric equations Page 12
Figure 3
The curve C shown in Figure 3 has parametric equations
x = t 3 – 8t, y = t 2 where t is a parameter. Given that the point A has parameter t = –1, (a) find the coordinates of A.
(1) The line l is the tangent to C at A. (b) Show that an equation for l is 2x – 5y – 9 = 0.
(5) The line l also intersects the curve at the point B. (c) Find the coordinates of B.
(6)
(C4 Jan 2009, Q7.)
C4 Maths Parametric equations Page 13
6.
Figure 2 Figure 2 shows a sketch of the curve with parametric equations
x = 2 cos 2t, y = 6 sin t, 0 t 2
.
(a) Find the gradient of the curve at the point where t = 3
.
(4)
(b) Find a Cartesian equation of the curve in the form
y = f(x), –k x k, Stating the value of the constant k.
(4)
(c) Write down the range of f(x). (2)
(C4 June 2009, Q5.)
C4 Maths Parametric equations Page 14
7. A curve C has parametric equations
x = sin2 t, y = 2 tan t , 0 ≤ t < 2
.
(a) Find x
y
d
d in terms of t.
(4)
The tangent to C at the point where t = 3
cuts the x-axis at the point P.
(b) Find the x-coordinate of P.
(6)
(C4 June 2010, Q4.)
8. The curve C has parametric equations
x = ln t, y = t2 −2, t > 0. Find (a) An equation of the normal to C at the point where t = 3,
(6)
(b) A Cartesian equation of C. (3)
(C4 Jan 2011, Q6.)
C4 Maths Parametric equations Page 15
9.
Figure 3 Figure 3 shows part of the curve C with parametric equations
x = tan , y = sin , 0 < 2
.
The point P lies on C and has coordinates
3
2
1,3 .
(a) Find the value of at the point P. (2)
The line l is a normal to C at P. The normal cuts the x-axis at the point Q.
(b) Show that Q has coordinates (k3, 0), giving the value of the constant k. (6)
(C4 June 2011, Q7.)
C4 Maths Parametric equations Page 16
10.
Figure 2 Figure 2 shows a sketch of the curve C with parametric equations
x = 4 sin
6
t , y = 3 cos 2t, 0 t < 2.
(a) Find an expression for x
y
d
d in terms of t.
(3)
(b) Find the coordinates of all the points on C where x
y
d
d = 0.
(5) (C4 Jan 2012, Q5.)
C4 Maths Parametric equations Page 17
11.
Figure 2 Figure 2 shows a sketch of the curve C with parametric equations
x = 3 sin 2t, y = 4 cos2 t, 0 t .
(a) Show that x
y
d
d = k3 tan 2t, where k is a constant to be determined.
(5)
(b) Find an equation of the tangent to C at the point where t = 3
.
Give your answer in the form y = ax + b, where a and b are constants.
(4)
(c) Find a Cartesian equation of C. (3)
(C4 June 2012, Q6.)
C4 Maths Parametric equations Page 18
12.
Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations
x = 1 – 2
1t, y = 2t – 1.
The curve crosses the y-axis at the point A and crosses the x-axis at the point B. (a) Show that A has coordinates (0, 3).
(2)
(b) Find the x-coordinate of the point B. (2)
(c) Find an equation of the normal to C at the point A. (5)
(C4 Jan 2013, part of Q5.)
C4 Maths Parametric equations Page 19
13. A curve C has parametric equations
x = 2sin t, y = 1 – cos 2t, 2
≤ t ≤
2
(a) Find d
d
y
x at the point where t =
6
.
(4)
(b) Find a cartesian equation for C in the form
y = f(x), –k ≤ x ≤ k, stating the value of the constant k.
(3)
(c) Write down the range of f(x). (2)
(C4 June 2013, Q4)
14.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
327secx t , 3tany t , 0 ≤ t ≤ 3
C4 Maths Parametric equations Page 20
(a) Find the gradient of the curve C at the point where t = 6
.
(4)
(b) Show that the cartesian equation of C may be written in the form
2 13 2( 9)y x , a ≤ x ≤ b
stating values of a and b. (3)
(C4 June 2013_R, part of Q7)
15.
Figure 3
Figure 3 shows a sketch of the curve C with parametric equations
4cos6
x t
, y = 2sin t, 0 ≤ t ≤ 2π
(a) Show that
x + y = 2√3 cos t
(3)
(b) Show that a cartesian equation of C is
(x + y)2 + ay2 = b
where a and b are integers to be determined.
(2)
(C4 June 2014, Q5)
C4 Maths Parametric equations Page 21
16.
Figure 3
The curve shown in Figure 3 has parametric equations
x = t – 4 sin t, y = 1 – 2 cos t, 2 2
3 3t
The point A, with coordinates (k, 1), lies on the curve.
Given that k > 0
(a) find the exact value of k,
(2)
(b) find the gradient of the curve at the point A.
(4)
There is one point on the curve where the gradient is equal to 1
2 .
(c) Find the value of t at this point, showing each step in your working and giving your answer to
4 decimal places.
[Solutions based entirely on graphical or numerical methods are not acceptable.]
(6)
(C4 June 2014_R, Q8)
C4 Maths Parametric equations Page 22
17. A curve C has parametric equations
x = 4t + 3, y = 4t + 8 + t2
5, t 0.
(a) Find the value of x
y
d
d at the point on C where t = 2, giving your answer as a fraction in its
simplest form.
(3)
(b) Show that the Cartesian equation of the curve C can be written in the form
y = 3
2
x
baxx, x 3,
where a and b are integers to be determined.
(3)
(C4 June 2015, Q5)