## Parametric Equations Exercise

1. Find the parametric equations for each of these circles:

 a x2 + y2 = 16 b x2 + y2 = 1 c (x + 3)2 + (y + 4)2 = 9 d (x − 1)2 + (y − 2)2 = 100 e x2 + (y − 5)2 = 25 f (x − 6)2 + y2 = 81

2. Give the parametric equations for each parabola:

 a y2 = 12x b y2 = 40x c (y − 5)2 = 8x d (y + 2)2 = 12x e (y − 1)2 = 4(x − 1) f (y + 1)2 = 8(x − 2)

3. Give the parametric equations for these ellipses and hyperbolas:

 a b c d 4. Given each set of parametric equations, eliminate the parameter and form an equation in x and y only:

 a x = 5cos t , y = 4 sin t b x = 6cos θ , y = 2sin θ c x = cos t − 5, y = sin t − 3 d x = 2cos w -1, y = 7sin w + 3 e x = 0.25cos t, y = 0.2sin t f x = 2.5cos v, y = 0.5sin v

5. a. Find for each of the following pairs of parametric equations:

 a x = 3cos t − 3, y = 3sin t − 4 b x = 2t2 , y = 4t + 5 c x = 4cos t , y = 5sin t d x = 5sec t − 3 , y = 4tan t + 1

6. By eliminating t, find the Cartesian equation of the curve x = t − 3, y = 4t2

7. x = t + 3 and y = 2t + 5. Find the cartesian equation of this function and sketch its graph.

8. A curve is given parametrically by x = 5secθ and y = 3tan θ

a. Find the Cartesian equation of this curve.

b. Sketch the curve, marking clearly any intercepts and the point P where θ = 4π / 3

c. Find at the point P

9. A curve y = p(x) has parametric equations x = -1 / 2t and y = 4t

a. Eliminate t to find the Cartesian equation of the curve.

b. Find the gradient of the curve at the point when t = 3

c. For what values of t is this curve discontinuous?

10. Write down the parametric equations of the conic 