1019 Words5 Pages

MEI Core 4
Parametric Equations
Chapter Assessment
1. A curve is defined by the parametric equations x = 2t 2 , y = 4t . (i) By eliminating the parameter, find the cartesian equation of the curve. (ii) Find the equation of the tangent to the curve at the point A with parameter t = 2. (iii) Show that the tangent does not meets the curve again. (iv) The normal of the curve at A cuts the curve again at B. Find the coordinates of B. [3] [4] [3] [5]
2. Find the turning points of the curve with parametric equations x = 3t , y = 12t − t 3 and distinguish between them. [5] 3. A circle is defined by the parametric equations x = 1 + 2 cos θ , y = 3 + 2sin θ . (i) Sketch the circle. dy (ii) Find at the point with parameter θ . dx (iii) Find the equation of the tangent at the point with parameter θ . π (iv) Find the coordinates of the point where θ = . 3 π (v) Find the equation of the normal at the point where θ = . 3
[2] [3] [3] [2] [4]
4. A line is defined by the parametric equations x = cos 2t , y = sin 2 t dy . (i) Find dx (ii) Find the cartesian equation of the line.
[3] [3]
Total 40 marks
© MEI, 01/09/08
1/5
MEI C4 Parametric Equations Assessment solutions
Solutions to Chapter Assessment
1. (i)
x = 2t 2 y = 4t ⇒ t =
Insert into
y
4
:
2 y2 ⎛y⎞ x = 2⎜ ⎟ ⇒ x = 8 ⎝4⎠ 2 ⇒ y = 8x
(ii) x = 2t 2 ⇒
dx = 4t dt dy y = 4t ⇒ =4 dt dy dy / dt 4 1 = = = dx dx / dt 4t t dy 1 = When t = 2: dx 2
[3]
When t = 2: x = 2 × 2 2 = 8 and y = 4 × 2 = 8
1 So the tangent has gradient 2 and passes through (8, 8) 1 Equation of tangent is y − 8 = 2 ( x − 8 ) 2 y − 16 = x − 8 2y = x + 8
(iii) Where tangent meets curve, 2 × 4t = 2t + 8
2
[4]
2t 2 − 8t + 8 = 0
t 2 − 4t + 4 = 0
(t − 2)2 = 0 The only root is the repeated root at t = 2 (the point of the tangent) so the tangent doesn’t meet the curve again. [3] 1 (iv) Gradient of tangent =

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