6. Circles

Question 1

How to go from two points on a circle to the equation of the diameter

Answer 1

Find the perpendicular bisector

Question 2

How to find the intersection of a line and a circle

Answer 2

Substitute either x or y from the line in place of x and y in the circle equation and solve simultaneously

Question 3

Names of lines with 0/1/2 intersections with the circle

Answer 3

2- secant
1- tangent
0- doesn’t touch the circle

Use the discriminant to find out which

Question 4

Circle theorems to know:

Answer 4

Radius meets a tangent at 90 degrees

The perpendicular bisector of any chord passes through the centre of the circle (intersection of any two is the centre)

Question 5

If all 3 vertices of a triangle are on the circumference

Answer 5

The triangle inscribes the circle
The circle circumscribes the triangle
If the circumscribing shape is a circle it is known as the circumcircle
The centre of a circumcircle is known as the circumcentre

Question 6

If angle ABC = 90 in a circumscribed triangle

Answer 6

AC is the diameter of the circumcircle

Question 7

Finding the centre of a circumcircle from a triangle or the 3 points

Answer 7

Find the intersection of the perpendicular bisectors

Question 8

How to show that AB is a diameter of the circumcircle?

Answer 8

Show that AC^2 + BC^2 = AB^2

Or show that AC is perpendicular to BC

Question 9

Equation of circle centre (a,b), radius r

Answer 9

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

Question 10

How to go from x^2 + y^2 + ax + by + c =0 to the equation of the circle?

Answer 10

1. Complete the square for the x and y terms
2. Expand and minus the number that was made
3. Rearrange to find (x-a)^2 + (y-b)^2 = r^2
4. Centre = (a,b) radius = r