Coordinate Geometry - PowerPoint Presentation

Slide1

Coordinate Geometry

in the (x,y) planeSlide2

IntroductionThis chapter focuses on Parametric equationsParametric equations split a ‘Cartesian’ equation into an x and y ‘component’

They are used to model projectiles in PhysicsThis in turn allows video game programmers to create realistic physics engines in their games…Slide3

Teachings for Exercise 2ASlide4

Coordinate Geometry in the (x,y) planeYou 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:for2A 

t

x

= 2t

y

= t

2

-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 valuesSlide5

Coordinate Geometry in the (x,y) planeYou can define x and y coordinates on a curve using separate functions, x = f(t) and y = g(t)

A curve has parametric equations:Find the Cartesian equation of the curve2A 

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

Divide by 2

Replace t with

x

/

2

SimplifySlide6

Coordinate Geometry in the (x,y) planeYou can define x and y coordinates on a curve using separate functions, x = f(t) and y = g(t)

A curve has parametric equations:Find the Cartesian equation of the curve2A 

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

Multiply by (t + 1)

Divide by x

Subtract 1

Replace t

1 =

x

/

x

Put the fractions together

S

implifySlide7

Teachings for Exercise 2BSlide8

You need to be able to use Parametric equations to solve problems in coordinate geometryThe 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.

2BCoordinate Geometry in the (x,y) plane

A

B

At points A and B, y = 0



Sub y = 0 in to find t at these points

y

= 0

+ t

2

Remember both answers

So t = ±2



Sub this into the x equation to find the x-coordinates of A and B

Sub in t = 2

Sub in t = -2



A and B are (-3,0) and (1,0)Slide9

You need to be able to use Parametric equations to solve problems in coordinate geometryA curve has Parametric equations:where a is a constant. Given that the curve passes through (2,0), find the value of a.

2BCoordinate Geometry in the (x,y) plane

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

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

Sub in x and t values

Divide by -2

The actual equations with a = -1Slide10

You need to be able to use Parametric equations to solve problems in coordinate geometryA curve is given Parametrically by:The line x + y + 4 = 0 meets the curve at point A. Find the coordinates of A.

2BCoordinate Geometry in the (x,y) plane

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

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…

Sub t = -2

Sub t = -2

The curve and line meet at (4, -8)Slide11

Teachings for Exercise 2CSlide12

You need to be able to convert Parametric equations into Cartesian equationsA Cartesian equation is just an equation of a line where the variables used are x and y onlyA curve has Parametric equations:

a) Find the Cartesian equation of the curveb) Sketch the curve2CCoordinate Geometry in the (x,y) plane

We need to eliminate t from the equations

Isolate

sint

Isolate cost

Replace

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 = 1Slide13

You need to be able to convert Parametric equations into Cartesian equationsA Cartesian equation is just an equation of a line where the variables used are x and y onlyA curve has Parametric equations:

a) Find the Cartesian equation of the curve2CCoordinate Geometry in the (x,y) plane

We need to eliminate t from the equations

Use the double angle formula from C3

Replace

sint

with x

Square

Rearrange

Replace sin

2

t with x

2

Square root

We can now replace

cos

t

Another way of writing this (by squaring the whole of each side)Slide14

Teachings for Exercise 2DSlide15

You need to be able to find the area under a curve when it is given by Parametric equationsThe Area under a curve is given by:

By the Chain rule:2DCoordinate Geometry in the (x,y) plane

Integrate the equation with respect to x

y multiplied by

dx

/

dt

(x

d

ifferentiated 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!)Slide16

You need to be able to find the area under a curve when it is given by Parametric equationsA curve has Parametric equations:Work out:

2DCoordinate Geometry in the (x,y) plane

Differentiate

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!Slide17

You need to be able to find the area under a curve when it is given by Parametric equationsThe diagram shows a sketch of the curve with Parametric equations:

The curve meets the x-axis at x = 0 and x = 9. The shaded region is bounded by the curve and the x-axis.Find the value of t when:x = 0x = 9b) Find the Area of R2DCoordinate Geometry in the (x,y) plane

0

9

R

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 Slide18

You need to be able to find the area under a curve when it is given by Parametric equationsThe diagram shows a sketch of the curve with Parametric equations:

The curve meets the x-axis at x = 0 and x = 9. The shaded region is bounded by the curve and the x-axis.Find the value of t when:x = 0x = 9b) Find the Area of R Limits of t are 3 and 02DCoordinate Geometry in the (x,y) plane

0

9

R

Differentiate

Sub in y and

dx

/

dt

Multiply

INTEGRATE!! (don’t forget!)

Sub in 3 and 0

Work out the answerSlide19

You need to be able to find the area under a curve when it is given by Parametric equationsThe diagram shows a sketch of the curve with Parametric equations:

Calculate the finite area inside the loop…2DCoordinate Geometry in the (x,y) plane

0

8

We have the x limits, we need the t limits

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)Slide20

You need to be able to find the area under a curve when it is given by Parametric equationsThe diagram shows a sketch of the curve with Parametric equations:

Calculate the finite area inside the loop…2DCoordinate Geometry in the (x,y) plane

The t limits are 0 and ±2

Sub in y and

dx

/

dt

Multiply out

INTEGRATE!!!!

Sub in

2

and

0

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

8Slide21

You need to be able to find the area under a curve when it is given by Parametric equationsThe diagram shows a sketch of the curve with Parametric equations:

Calculate the finite area inside the loop…2DCoordinate Geometry in the (x,y) plane

Sub in y and

dx

/

dt

Multiply out

INTEGRATE!!!!

Sub in 0 and -2

Differentiate

Here you end up subtracting a negative…

Double

The t limits are 0 and ±2

0

8Slide22

SummaryWe have learnt what Parametric equations areWe have seen how to write them as a Cartesian equationWe have seen how to calculate areas beneath

Parametric equations