Coordinate Geometry in the x y plane After

Coordinate Geometry in the (x, y) plane

After completing this chapter you should be able to: • Sketch the graph of a curve given it’s parametric equation • Use the parametric equation of the curve to solve various problems including the intersection of a line with a curve • Convert parametric equations into a Cartesian form • Find the area under a curve whose equation is expressed in parametric form

2. 1 You can define the coordinates of a point on a curve using parametric equations. In parametric equations, coordinates x and y are expressed as x = f(t) and y = g(t), where the variable t is a parameter. Sweet little things, they involve a three way relationship with x, y and t. Something like x = 2 t and y = t - 4 for 0 ≤ t ≤ 6 So you pop the values for t into the equations for x and y to give you some coordinates to plot.

t x = 2 t y = t-4 0 0 -4 1 2 -3 Which gives this graph 2 4 -2 3 6 -1 4 8 0 5 10 1 6 12 2

Exercise 2 A page 12

2. 2 You need to be able to use parametric equations to solve problems in coordinate geometry The first problem is when a curve meets the X or Y axis and you need to find the coordinates of the intersection Lets imagine the curve with parametric equations x = t², y = (1 -t)(t+3) meets the x axis when y = 0 then t = 1 giving x = 1 or t = -3 so x = 9 the coordinates of the intersections are (1, 0) and (9, 0)

The second problem occurs when one of the parametric equations has an unknown in it you need to find. a curve has parametric equations x = at², y = a(8 t³- 1) and passes through the point (4, 0). Find the value of a. using this value of t gives a = 16

The third type of problem occurs when a line and a curve meet and you need to find the point of intersection. The line x + y = 9 meets the curve with parametric equations x = t², y= (t+3)(t-2) at points P and Q. Find the coordinates of P and Q

using t = -3 gives us x = 9 and y = 0

2. 3 You need to be able to convert parametric equations into a Cartesian equation You can find the Cartesian equation by eliminating the t

Let’s find the Cartesian equation of the ellipse we looked at earlier x = 2 – 4 cosϴ, y = 3 sinϴ + 4, 0 ≤ ϴ ≤ 2Π now we combine these two results using cos²ϴ + sin²ϴ = 1

exercise 2 C page 17

2. 4 You need to be able to find the area under a curve given by parametric equations

lets find the area of the finite region enclosed by the loop of the curve with parametric equations x = t² - 1 , y = ½(t – t)³ -2 ≤ t ≤ 2 limits for integration when x = 0 t = ± 1 this gives us turning to the next slide we see

exercise 2 D page 20