Coordinate Geometry

Question 1

What is the formula for the gradient between two points?

Answer 1

m = (y2 - y1) / (x2 - x1)

Question 2

How to make sure that two lines are perpendicular?

Answer 2

their gradients x to make -1
or
one is the negative reciprocal of the other

Question 3

How to find the midpoint of the line joining two points?
How to find the length?

Answer 3

M = ( x1 + x2 / 2 , y2 + y1 / 2 )

for length - Pythagoras

Question 4

What are the forms of a straight line?

Answer 4

Y = mx + c
or
ax + by + c = 0 where a, b and c are integers

Question 5

What should you include when sketching graphs?

Answer 5

y - intercept
Any roots
Turning point (if quadratic)
Gradient (if linear)

Question 6

what is the alternate equation of a line when given a gradient and a point it passes through?

Answer 6

y - y1 = m( x - x1)

where m is the gradient and (x1 , y1) is the point.

Question 7

What is the equation of the line when given two points?

Answer 7

1) calculate gradient
2) input gradient and one point into y - y1 = m( x - x1)

y - y1 = y2 - y1 / x2 - x1 (x - x1)

Question 8

What is the equation for a circle?

Answer 8

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

Question 9

How do you find the equation of a circle given:
1) Its centre and radius?
2) Its centre and a point on the outside?

Answer 9

1) just input a, b & r from centre (a,b) and radius r into (x - a)^2 + (y - b)^2 = r^2

2) use pythagoras to find the radius then us this and centre a, b to substitute into equation

Question 10

What are the 3 circle theormes to know relating to circles?

Answer 10

1) The angle in the semi circle is a right angle ( If A, B and C are points on a circle and AB is the diameter oof the circle, then ABC = 90 degrees)
2) the perpendicular bisector of a chord passes through the centre of the circle
3) the tangent and radius meet at 90 degrees - perpendicular

Question 11

How do you find the equation of a circle given 3 points on its circumfrence?

Answer 11

1) all points on circumfrence must be equidistant to the centre. Perpendicular bisector of AB must pass through the centre as equidistant from A & B. Same for AC
Hence, you must first calculate the perpendicular bisector of any two AB, BC or AC using gradients and midpoints

2) solve these two bisectors simultaneously to give the centre of the circle (where the two bisectors meet)

3) Finally, use this and any point (A, B, C) to find radius of the circle and hence find the equation of the circle.

Question 12

How do you find the point of intersection between a line and a circle?

Answer 12

Solve simultaneously with substitution

when finding the the point(s) of intersection between 2 lines, solve simultaneously. It is always useful to equate for y first and then solve this way.

Question 13

How do you know if two lines are parallel?

Answer 13

They have the same gradient

Question 14

How do you calculate the area of any triangle?

Answer 14

1/2 x base x perpendicular height

Question 15

How do you work out where two lines intersect?

Answer 15

Solve simultaneously

Question 16

If A, B and C are points which lie one a circle, how do you show that AB is a diameter of the circle?

Answer 16

Show that AB^2 + BC^2 = AB^2
or
find the gradients of AC and BC and show that AC is perpendicular to BC

Question 17

Given three points A, B and C, which lie on a circle, how do you find the centre of the circle?

Answer 17

Find the equation of perpendicular bisectors of two sides of the sides AB, AC or BC
Find the point of intersection of those two perpendicular bisectors

Question 18

How do you prove a straight line and a circle will never intersect?

Answer 18

Try to solve simultaneously to produce a quadratic equation

show the value of the discriminant is negative

Question 19

If you are given points A and B where AB is the diameter of the circle, how do you find the equation of the circle?

Answer 19

• Find midpoint of AB, gives circle centre
• Find length of AB which gives diameter
• Half the length to find the radius
• Use (x - a)^2 + (y - b)^2 = r^2