Edexcel Core Mathematics 4 Parametric equations. Edited by: K V Kumaran kumarmaths.weebly.com 1
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 = t + t 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 sin θ + cos 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. kumarmaths.weebly.com
kumarmaths.weebly.com 3
Example The curve C is described by the parametric equations x = 5cost, y = cost, 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: Therefore: cost cos t sin t cos t 1 y = cos t 1 If x = 5cost then cost = x 5 So finally y = x - 1 5 b) This is simply a quadratic that is symmetrical about the y axis and intercepts with the y axis at -1. -1 kumarmaths.weebly.com 4
kumarmaths.weebly.com 5
Example The parametric equations of a curve are x = 14 sin t, y = 14t cos t π dy where 0 < t <. Find in terms of t, and hence show that the gradient of the curve is zero where tan t 1 t dy Since the curve is given parametrically we can use the chain rule to find dy dy dt dt So by differentiating the parametric: x = 14 sin t dy = 14 cos t dt y = 14t cos t (a Product) = 14 cos 14t sin t dt dy dy dt dt dy 14 cost - 14t sin t 1 ttant 14cost When the gradient is zero 1 ttan t 0 tan t 1 t This equation could be solved by iterative methods (C3). kumarmaths.weebly.com 6
Example 4 A curve is given by the parametric equations x = 7 sin 3 t, y = 6 cos t, 0 < t < 4 Show that dy By chain rule 7 sin t 8 dt dy dt dy 1sin tcost dt dy 1sint don't forget the. dt By C3 trig identities sin t = sin t cos t dt 1sin t cos t dy dt dy 4sin tcost 7 sin t 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 t t 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. kumarmaths.weebly.com 7
a) Find the gradient of the curve at the point P where t= 3 Using chain rule: dy sec t cost dt dt dt dy dt dy Let t= 3 π dy cos t cos tcost sec t Grad = cos cos 0. 5 3 3 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 3 respectively. Using y = mx + c 3 = 4 3 + c c = 7 3 Therefore the equation of the normal is y 4x 7 3 kumarmaths.weebly.com 8
Differentiating a x This function describes growth and decay, and its derivative gives a measure of the rate of change of this growth/decay. Since y = a x, taking logs of both sides gives ln y = ln a x = x ln a. Using implicit differentiation to differentiate ln y: 1 dy = ln a y dy = y ln a = ax ln a This result needs to be learn, and is not given in the formula sheet. kumarmaths.weebly.com 9
C4 Parametric differentiation past paper questions 1. A curve has parametric equations x = cot t, y = sin t, 0 < t (a) Find an expression for dy 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 005, Q6.). Figure y 0.5 1 0.5 O 0.5 1 x The curve shown in Figure has parametric equations x = sin t, y = sin t, 6 < t <. (a) Find an equation of the tangent to the curve at the point where t = 6 (b) Show that a cartesian equation of the curve is y = 3 1 x + (1 x ), 1 < x < 1. (6) (C4 June 006, Q4.) kumarmaths.weebly.com 10
3. A curve has parametric equations x = 7 cos t cos 7t, y = 7 sin t sin 7t, < t <. 8 3 dy (a) Find an expression for in terms of t. You need not simplify your answer. (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 007, Q3.) 4. A curve has parametric equations (a) Find an expression for x = tan t, y = sin t, 0 < t < dy in terms of t. You need not simplify your answer. (b) Find an equation of the tangent to the curve at the point where t = 4 5. 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 y = f(x). (4) (C4 June 007, Q6.) Figure 3 The curve C shown in Figure 3 has parametric equations x = t 3 8t, y = t 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 x 5y 9 = 0. (5) The line l also intersects the curve at the point B. (c) Find the coordinates of B. (6) (C4 Jan 009, Q7.) kumarmaths.weebly.com 11
6. Figure Figure shows a sketch of the curve with parametric equations x = cos t, y = 6 sin t, 0 t (a) Find the gradient of the curve at the point where t = 3 (b) Find a Cartesian equation of the curve in the form (4) Stating the value of the constant k. (c) Write down the range of f(x). 7. A curve C has parametric equations y = f(x), k x k, (4) () (C4 June 009, Q5.) x = sin t, y = tan t, 0 t < dy (a) Find in terms of t. The tangent to C at the point where t = cuts the x-axis at the point P. 3 (b) Find the x-coordinate of P. (4) (6) (C4 June 010, Q4.) kumarmaths.weebly.com 1
8. The curve C has parametric equations Find x = ln t, y = t, t > 0. (a) An equation of the normal to C at the point where t = 3, (b) A Cartesian equation of C. (6) (C4 Jan 011, Q6.) 9. Figure 3 Figure 3 shows part of the curve C with parametric equations x = tan, y = sin, 0 < 1 The point P lies on C and has coordinates 3, 3 (a) Find the value of at the point P. () 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 (k 3, 0), giving the value of the constant k. (6) (C4 June 011, Q7.) kumarmaths.weebly.com 13
10. Figure Figure shows a sketch of the curve C with parametric equations 11. x = 4 sin t, 6 y = 3 cos t, 0 t <. dy (a) Find an expression for in terms of t. d y (b) Find the coordinates of all the points on C where = 0. (5) (C4 Jan 01, Q5.) Figure kumarmaths.weebly.com 14
Figure shows a sketch of the curve C with parametric equations x = 3 sin t, y = 4 cos t, 0 t. (a) Show that d y = k 3 tan t, 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. (c) Find a Cartesian equation of C. (4) 1. (C4 June 01, Q6.) Figure Figure shows a sketch of part of the curve C with parametric equations x = 1 1 t, y = t 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). (b) Find the x-coordinate of the point B. (c) Find an equation of the normal to C at the point A. () () (5) (C4 Jan 013, part of Q5.) kumarmaths.weebly.com 15
13. A curve C has parametric equations x = sin t, y = 1 cos t, t (a) Find d y at the point where t = 6 (b) Find a cartesian equation for C in the form (4) y = f(x), k x k, 14. stating the value of the constant k. (c) Write down the range of f(x). () (C4 June 013, Q4) Figure Figure shows a sketch of the curve C with parametric equations 3 x 7sec t, y 3tan t, 0 t 3 (a) Find the gradient of the curve C at the point where t = 6 (b) Show that the cartesian equation of C may be written in the form (4) stating values of a and b. 1 3 y ( x 9), a x b (C4 June 013_R, part of Q7) kumarmaths.weebly.com 16
15. Figure 3 Figure 3 shows a sketch of the curve C with parametric equations x 4cos t, y = sin t, 0 t π 6 (a) Show that x + y = 3 cos t (b) Show that a cartesian equation of C is (x + y) + ay = b 16. where a and b are integers to be determined. () (C4 June 014, Q5) Figure 3 kumarmaths.weebly.com 17
The curve shown in Figure 3 has parametric equations x = t 4 sin t, y = 1 cos t, t 3 3 The point A, with coordinates (k, 1), lies on the curve. Given that k > 0 (a) find the exact value of k, (b) find the gradient of the curve at the point A. There is one point on the curve where the gradient is equal to 1 () (4) (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 014_R, Q8) 17. A curve C has parametric equations x = 4t + 3, y = 4t + 8 + 5, t 0. t (a) Find the value of form. dy at the point on C where t =, giving your answer as a fraction in its simplest (b) Show that the Cartesian equation of the curve C can be written in the form y = x ax b, x 3, x 3 where a and b are integers to be determined. (C4 June 015, Q5) kumarmaths.weebly.com 18
18. Figure shows a sketch of the curve C with parametric equations x = 4 tan t, y = 5 3 sint, 0 t < 15 The point P lies on C and has coordinates 4 3, (a) Find the exact value of d y at the point P. Give your answer as a simplified surd. (4) The point Q lies on the curve C, where d y = 0. (b) Find the exact coordinates of the point Q. () (C4 June 016, Q5) 19. The curve C has parametric equations x = 3t 4, y = 5 6 t, t > 0 (a) Find dy in terms of t () The point P lies on C where t = 1 (b) Find the equation of the tangent to C at the point P. Give your answer in the form y = px + q, where p and q are integers to be determined. (c) Show that the cartesian equation for C can be written in the form y = ax + b x + 4, x > 4 where a and b are integers to be determined. (C4 June 017, Q1) kumarmaths.weebly.com 19
0. The curve C has parametric equations x = 10 cos t, y = 6 sin t, The point A with coordinates (5, 3) lies on C. t 1. (a) Find the value of t at the point A. (b) Show that an equation of the normal to C at A is 3y = 10x 41 The normal to C at A cuts C again at the point B. (c) Find the exact coordinates of B. (1) (6) (8) (IAL C34 June 014, Q11) Figure 3 shows a sketch of part of the curve with parametric equations x = t + t, y = t 3 9t, t The curve cuts the x-axis at the origin and at the points A and B as shown in Figure 3. (a) Find the coordinates of point A and show that point B has coordinates (15, 0). (b) Show that the equation of the tangent to the curve at B is 9x 4y 135 = 0. (5) The tangent to the curve at B cuts the curve again at the point X. (c) Find the coordinates of X. (5) (IAL C34 June 015, Q9) kumarmaths.weebly.com 0
DO NOT WRITE IN THIS AREA. A curve C has parametric equations (a) Show that d y x = 6 cos t, y = sin t, t = λ cosec t, giving the exact value of the constant λ. (4) (b) Find an equation of the normal to C at the point where t 3 Give your answer in the form y = mx + c, where m and c are simplified surds. (6) The cartesian equation for the curve C can be written in the form x = f(y), k < y < k where f(y) is a polynomial in y and k is a constant. 3. (c) Find f(y). (d) State the value of k. (1) (IAL C34 Jan 016, Q9) 13. y C A B Figure 4 The curve C shown in Figure 4 has parametric equations O x x = 1+ 3tanq, y = 5secq, - p < q < p The curve C crosses the y-axis at A and has a minimum turning point at B, as shown in Figure 4. (a) Find the exact coordinates of A. (b) Show that dy = l sinq, giving the exact value of the constant l. (4) (c) Find the coordinates of B. () (d) Show that the cartesian equation for the curve C can be written in the form y k x x 4 where k is a simplified surd to be found. (IAL C34 Jan 017, Q13) kumarmaths.weebly.com 1
DO NOT WRITE IN THIS AR 4. 14. y l P S C O Figure 6 x Figure 6 shows a sketch of the curve C with parametric equations x = 8cos 3 θ, y = 6sin θ, 0 θ p Given that the point P lies on C and has parameter θ = p 3 (a) find the coordinates of P. The line l is the normal to C at P. (b) Show that an equation of l is y = x + 3.5 () (5) (IAL C34 June 017, Q14) kumarmaths.weebly.com