introduction this chapter focuses on parametric equations parametric equations split a...

Coordinate Geometryin the (x,y) plane

Introduction

• This chapter focuses on Parametric equations

• Parametric equations split a ‘Cartesian’ equation into an x and y ‘component’

• They are used to model projectiles in Physics

• This in turn allows video game programmers to create realistic physics engines in their games…

Teachings for Exercise 2A

Coordinate Geometry in the (x,y) plane

You can define x and y coordinates on a curve using

separate functions, x = f(t) and y = g(t)

The letter t is used as the primary application of this is in Physics, with

t representing time.

Draw the curve given by the Parametric Equations:

for

2A

𝑥=2 𝑡 𝑦=𝑡 2

−3 ≤𝑡 2≤3

t

x = 2t

y = t2

-3

-6

9

-2

-4

4

-1

-2

1

0

0

0

1

2

1

2

4

4

3

6

9

10-10

10

-10

Work out the x and y co-ordinates to plot by substituting

the t values




A curve has parametric equations:

Find the Cartesian equation of the curve

2A

𝑥=2 𝑡𝑦=𝑡 2

Rewrite t in terms of x, and then substitute it into the y

equation…

𝑥=2 𝑡𝑥2=𝑡

𝑦=𝑡 2

𝑦=(𝑥2 )2

𝑦= 𝑥2

4

Divide by 2

Replace t with x/2

Simplify




A curve has parametric equations:

Find the Cartesian equation of the curve

2A

𝑥=1𝑡+1 𝑦=𝑡 2

Rewrite t in terms of x, and then substitute it into the y

equation…

Multiply by (t + 1)

𝑥=1𝑡+1

𝑥 (𝑡+1)=1

𝑡+1=1𝑥

𝑡=1𝑥−1

Divide by x

Subtract 1

𝑦=𝑡 2

𝑦=( 1𝑥 −1)2

𝑦=( 1𝑥 − 𝑥𝑥 )2

𝑦=( 1−𝑥𝑥 )2

𝑦=(1−𝑥)𝑥2

2

Replace t

1 = x/x

Put the fractions together

Simplify

Teachings for Exercise 2B

You need to be able to use Parametric equations to solve

problems in coordinate geometry

The diagram shows a sketch of the curve with Parametric equations:

The curve meets the x-axis at the points A and B. Find their coordinates.

2B


𝑥=𝑡−1 𝑦=4−𝑡 2 A B

At points A and B, y = 0 Sub y = 0 in to find t at these

points

𝑦=4−𝑡 2

0=4−𝑡 2

𝑡 2=4𝑡=±2

y = 0

+ t2

Remember both answers

So t = ±2 Sub this into the x equation to find the x-coordinates of A and B 𝑥=𝑡−1

𝑥=(2)−1 𝑥=(−2)−1𝑥=1 𝑥=−3

𝑥=𝑡−1Sub in t =

2Sub in t = -

2 A and B are (-3,0) and (1,0)



A curve has Parametric equations:

where a is a constant. Given that the curve passes through (2,0), find the

value of a.

2B


𝑥=𝑎𝑡 𝑦=𝑎(𝑡 3+8)

We know there is a coordinate where x = 2 and y = 0, Sub y

= 0 into its equation

𝑦=𝑎(𝑡 3+8)

0=𝑎(𝑡 3+8)

𝑡 3+8=0𝑡 3=−8𝑡=−2

y = 0

The ‘a’ part cannot be 0 or there would not be any equations!

Subtract 8

Cube root

So the t value at the coordinate (2,0) is -2

Since we know that at (2,0), x = 2 and t = -2, we can put these into the x equation to find a

𝑥=𝑎𝑡(2)=𝑎(−2)−1=𝑎

Sub in x and t values

Divide by -2

𝑥=𝑎𝑡 𝑦=𝑎(𝑡 3+8)

𝑥=−𝑡 𝑦=−(𝑡 3+8)The actual equations

with a = -1



A curve is given Parametrically by:

The line x + y + 4 = 0 meets the curve at point A. Find the coordinates

of A.

2B


𝑥=𝑡 2 𝑦=4 𝑡

The first thing we need to do is to find the value of t at

coordinate A Sub x and y equations into

the line equation

𝑥=𝑡 2 𝑦=4 𝑡

𝑥+𝑦+4=0(𝑡 2)+(4 𝑡)+4=0

𝑡 2+4 𝑡+4=0(𝑡+2)2=0

𝑡=−2

Replace x and y with their equations

‘Tidy up’

Factorise

Find t

So t = -2 where the curve and line meet (point A)

We know an equation for x and one for y, and we now have the t value to put into

them…

𝑥=𝑡 2 𝑦=4 𝑡𝑥=(−2)2

𝑥=4𝑦=4 (−2)𝑦=−8

Sub t = -2

Sub t = -2

The curve and line meet at (4, -8)

Teachings for Exercise 2C

You need to be able to convert Parametric equations into

Cartesian equations

A Cartesian equation is just an equation of a line where the

variables used are x and y only


a) Find the Cartesian equation of the curve

b) Sketch the curve

2C


𝑥=𝑠𝑖𝑛𝑡+2𝑦=𝑐𝑜𝑠𝑡−3

𝑥=𝑠𝑖𝑛𝑡+2 𝑦=𝑐𝑜𝑠𝑡−3

We need to eliminate t from the equations

𝑥−2=𝑠𝑖𝑛𝑡 𝑦+3=𝑐𝑜𝑠𝑡Isolate

sintIsolate cost

𝑠𝑖𝑛2𝑡+𝑐𝑜𝑠2𝑡≡1

¿¿+¿ ¿1Replace sint and

cost

The equation is that of a circle

Think about where the centre will be, and its radius

5-5

5

-5

¿Centre = (2, -3)

Radius = 1

You need to be able to convert Parametric equations into

Cartesian equations

A Cartesian equation is just an equation of a line where the

variables used are x and y only


a) Find the Cartesian equation of the curve

2C


𝑥=𝑠𝑖𝑛𝑡 𝑦=𝑠𝑖𝑛2 𝑡

We need to eliminate t from the equations

𝑥=𝑠𝑖𝑛𝑡 𝑦=𝑠𝑖𝑛2 𝑡𝑦=2 𝑠𝑖𝑛𝑡𝑐𝑜𝑠𝑡𝑦=2 𝑥𝑐𝑜𝑠𝑡

𝑠𝑖𝑛2𝑡+𝑐𝑜𝑠2𝑡≡1𝑐𝑜𝑠2𝑡=1−𝑠𝑖𝑛2𝑡𝑐𝑜𝑠2𝑡=1−𝑥2

𝑐𝑜𝑠𝑡=√1− 𝑥2

𝑦=2 𝑥𝑐𝑜𝑠𝑡

𝑦=2 𝑥√1− 𝑥2

𝑦 2=4 𝑥2(1−𝑥2)

Use the double angle formula from

C3𝑥2=𝑠𝑖𝑛2𝑡Replace sint with x

Square

Rearrange

Replace sin2t with x2

Square root

We can now replace cos t

Another way of writing this (by squaring the whole of

each side)

Teachings for Exercise 2D

You need to be able to find the area under a curve when it is

given by Parametric equations

The Area under a curve is given by:

By the Chain rule:

2D


∫ 𝑦 𝑑𝑥

∫ 𝑦𝑑𝑥𝑑𝑡

𝑑𝑡

Integrate the equation with respect

to x

y multiplied by dx/dt (x differentiated with

respect to t)

Integrate the whole expression with respect to t

(Remember you don’t have to do anything with the dt at the end!)




Work out:

2D

Coordinate Geometry in the (x,y) plane∫ 𝑦

𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=5 𝑡2 𝑦=𝑡 3

∫1

2

𝑦 𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=5 𝑡2 𝑦=𝑡 3

𝑑𝑥𝑑𝑡

=10 𝑡Differentiate

∫1

2

𝑡 3(10 𝑡)𝑑𝑡

∫1

2

10 𝑡 4 𝑑𝑡

∫1

2


𝑑𝑡

¿ [2 𝑡5 ]12¿ [ 10 𝑡 55 ]

1

2

¿¿−¿¿ (64 )− (2 )¿62

Sub in y and dx/dt

Multiply

Remember to Integrate now. A

common mistake is forgetting to!

Sub in the two limits

Work out the answer!




The curve meets the x-axis at x = 0 and x = 9. The shaded region is bounded by the curve and the x-

axis.

a) Find the value of t when:i) x = 0ii) x = 9

b) Find the Area of R

2D


𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=𝑡 2 𝑦=2 𝑡(3−𝑡)𝑡≥00 9

R

𝑥=𝑡 2

0=𝑡 2

0=𝑡

𝑥=𝑡 2

9=𝑡 2

±3=𝑡3=𝑡

x = 0

Square root

x = 9

Square root

t ≥ 0

i) ii)

Normally when you integrate to find an area, you use the limits of x, and substitute them into the equation

When integrating using Parametric Equations, you need to use the limits of t (since we integrate with respect to t, not x)

The limits of t are worked out using the limits of x, as we have just done




The curve meets the x-axis at x = 0 and x = 9. The shaded region is bounded by the curve and the x-

axis.

a) Find the value of t when:i) x = 0ii) x = 9

b) Find the Area of R Limits of t are 3 and 0

2D


𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=𝑡 2 𝑦=2 𝑡(3−𝑡)𝑡≥00 9

R𝑥=𝑡 2𝑦=2 𝑡(3−𝑡)

𝑑𝑥𝑑𝑡

=2 𝑡Differentiate

∫0

3


𝑑𝑡

∫0

3

2𝑡 (3− 𝑡 ) (2𝑡 )𝑑𝑡

∫0

3

12 𝑡 2−4 𝑡 3𝑑𝑡

¿ [ 12 𝑡33 −4 𝑡 4

4 ]0

3

¿ [ 4 𝑡 3−𝑡 4 ]03

¿ (4 (3)3−(3)4 )− (4 (0)3−(0)4 )

¿ (108−81 )

¿27

Sub in y and dx/dt

Multiply

INTEGRATE!! (don’t forget!)

Sub in 3 and 0

Work out the answer




Calculate the finite area inside the loop…

2D


𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=2 𝑡2𝑦=𝑡 (4−𝑡 2)

0 8

We have the x limits, we need the t limits

𝑥=2 𝑡2

0=2 𝑡 2

0=𝑡

𝑥=2 𝑡2

8=2 𝑡 2

±2=𝑡

x = 0

Halve and square root

x = 8

Halve and square root

Our t values are 0 and ±2

We have 3 t values. We now have to integrate twice, once using 2 and once using -2

One gives the area above the x-axis, and the other the area below

(in this case the areas are equal but only since the graph is symmetrical)





2D


𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=2 𝑡2𝑦=𝑡 (4−𝑡 2)

The t limits are 0 and ±2

∫0

2


𝑑𝑡

𝑥=2 𝑡2 𝑦=𝑡 (4−𝑡 2)𝑑𝑥𝑑𝑡

=4 𝑡

∫0

2

𝑡 (4−𝑡 2) (4 𝑡)𝑑𝑡

∫0

2

16 𝑡 2−4 𝑡4 𝑑𝑡

¿ [ 16 𝑡 33 −4 𝑡5

5 ]0

2

¿ ( 16(2)33−4 (2)5

5 )−( 16(0)33−4(0)5

5 )¿17

115

Sub in y and dx/dt

Multiply out

INTEGRATE!!!!

Sub in 0 and -2

Differentiate

At this point we can just double the answer, but just to show you the other pairs give the same

answer (as the graph was symmetrical)

0 8





2D


𝑑𝑥𝑑𝑡

𝑑𝑡

𝑥=2 𝑡2𝑦=𝑡 (4−𝑡 2)

∫−2

0


𝑑𝑡

𝑥=2 𝑡2 𝑦=𝑡 (4−𝑡 2)𝑑𝑥𝑑𝑡

=4 𝑡

∫−2

0

𝑡 (4−𝑡2) (4 𝑡)𝑑𝑡

∫−2

0

16 𝑡 2−4 𝑡 4 𝑑𝑡

¿ [ 16 𝑡 33 −4 𝑡5

5 ]− 2

0

¿ ( 16(0)33−4 (0)5

5 )−( 16(−2)33−4 (−2)5

5 )¿17

115

Sub in y and dx/dt

Multiply out

INTEGRATE!!!!

Sub in 0 and -2

Differentiate

Here you end up subtracting a negative…

¿34215

Double

The t limits are 0 and ±2

0 8

Summary

• We have learnt what Parametric equations are

• We have seen how to write them as a Cartesian equation

• We have seen how to calculate areas beneath Parametric equations

introduction this chapter focuses on parametric equations parametric equations split a...

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