CIRCLES - alevel maths

Question 1

How do convert from circle equation form: (x-a)^2 +(y-b)^2=r^2
to form: x^2+y^2 +2fx +2gy+c=0

how do you convert back to the original form?

Answer 1

1. expand

2. Factorise by completing the square

Question 2

Complete the square for this equation

x^2 - 4x + y^2 - 6y -3

Answer 2

1. First deal with x’s then with y’s.
2. half the x term to get G, and put it in a bracket: (x - G)^2
3. square the G and minus it outside the bracket: (x-G)^2 -G^2
4. Repeat for the y
5. Add the two outcomes together

(x-2)^2 + (y-3) - 16

Question 3

Equation for mid point of a line, when you have two co-ordinates of the line (x1 , y1) & (x2 , y2)

Answer 3

the co-ordinates of the lines mid point will be
x co-ordinate= (x1+x2)/2
y co-ordinate=(y1+y2)/2

Question 4

gradient of a line equation?

when you have two points on the line

Answer 4

(y2 - y1) / (x2 - x1)

always do the same order of minusing so that you get the correct gradient sign. (you can always check it is correct by looking at the slope!)

Question 5

What is a perpendicular bisector of a line segment?

Answer 5

A line that passes through the mid point of a line segment and is perpendicular to that segment

Question 6

What is the gradient of a perpendicular bisector or a line segment?

Answer 6

-1 / (line segment gradient)

Question 7

The tangent to the circle is _______ to the radius of the circle at the intersection point

the perpendicular bisector to the chord of the circle will pass through the _____ of the circle.

Answer 7

perpendicular

centre

Question 8

what is a circumcircle?

Answer 8

a circle that passes through the vertices of a triangle

Question 9

How do the sides of a circumcircle relate to the midpoint of the circle?

Answer 9

The perpendicular bisectors o the sides will pass through the centre of the circle

Question 10

What will the angle be in a triangle that is in a semi-circle

Answer 10

90 degrees

Question 11

What is the 3 point rule used for

Answer 11

with 3 points on a circle, you can make 2 chords, whose perpendicular bisectors will intersect at the centre of the circle.

you find the mid-point & gradient of the two chords, meaning you can find the gradient and a point on the bisectors. Use these values to get an equation for each bisector and then find where they intersect.