co-ordinate geometry

Question 1

equations of a line

Answer 1

y-y, = m( x-x, )
y = mx + c
ax + by + c = 0

Question 2

easiest way to find
y-y, = m( x-x, )

Answer 2

1) label ( x , y )
and ( x, , y, )
2) find gradient ( m )
3) substitute back into the equation
4) covert to one of the other forms if necessary
e.g to get y = mx + c put everything on the right except y

Question 3

line segment

Answer 3

• a line that goes through two points
• midpoint formula
• Pythagoras for the length
• perpendicular have reciprocal gradient

Question 4

equation of a circle

Answer 4

( x-a )^2 + ( y-b )^2 = r^2

a = x coordinate of the centre
b = y coordinate of the centre
r = radius

complete the square to get to this form

Question 5

3 properties of a circle

Answer 5

• the angle in a semicircle is always a right angle
• the perpendicular line from the centre to a chore bisects the chord (midpoint)
• a radius and a tangent to the same point meet at right angles

Question 6

gradient rule for perpendicular lines

Answer 6

remember that the radius and a tangent to the same point meet perpendicularly

• find the perpendicular equation/radius using this fact

Question 7

parametric equation definition

Answer 7

normally (X,Y) graphs are described using cartesian equations ( a single equation linking x and y )

parametric is where x and y are each separately defined in terms of a third variable called a parameter (t)

Question 8

using graphs and tables

Answer 8

work out x =x and y by substituting a range of t values
Then plot the cartesian coordinates on the axes as normal

Question 9

circles can be given by parametric equations too

Answer 9

a circle with centre (0,0) and r radius is defined by the parametric equations

x=rcosϑ
y=rsinϑ

if the centre is (a,b) the it is

x=rcosϑ + a
y=rsinϑ + b

Question 10

Parametric equations to find where graphs intersect axis

Answer 10

if your finding an x intersect then u make y = 0

use the parametric equation for y to find the values of t where it crosses the x axis ( factorise )

Question 11

Parametric equations to find where graphs intersect a line

Answer 11

sub the parametric equations to the equation of the line

rearrange and factorise to get t values

go back to the parametric equations and substitute the t value back in to find the intersect coordinates

Question 12

rearranging parametric equations to cartesian equations

Answer 12

1. make the parameter (t) the subject on one of the parametric equations then substitute it into the other.

or

2. if ur equation involves trig use trig identities to eliminate the parameter (t)

Question 13

rearranging parametric equations to cartesian equations with trig identities

Answer 13

If you try and make ϑ the subject it get’s messy, find a way to get x and y in terms of the same trig function

cos2ϑ = 1 - 2sin^2ϑ