Chapter 6 - Circles

Question 1

How do you find the midpoint of a line segment with both end coordinates?

(x1 , y1) and (x2 , y2).

Answer 1

(x1 + x2 , y1 + y2)
(——— ——— )
( 2 2 )

Question 2

What is the perpendicular bisector of a line AB?

Answer 2

It is the straight line that is perpendicular to the line AB and passes through the midpoint of AB.

Question 3

What is the equation of a circle with the centre

(0 , 0) and the radius r.

Answer 3

X^2 + y^2 = r^2.

Question 4

What is the equation of a circle with the centre

(a , b) and radius r?

Answer 4

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

Question 5

What is the centre and radius of a circle if it is written like:

x^2 + y^2 + 2ax + 2by + c = 0?

Answer 5

Centre=
(-a, -b)

Radius=
/a^2 + b^2 -c

Question 6

How many different ways can a line touch a circle and describe them.

Answer 6

It can touch the circle in 2 different ways. But there is one other possible line.

Just touching the circle at one point.

Touching the circle at two points.

Not touching the circle at all.

Question 7

What is special about a tangent to a circle?

Answer 7

It is perpendicular to the radius of the circle at the point of intersection.

Question 8

What is special about the perpendicular bisector of a chord?

Answer 8

It always goes through the centre of the circle.

Question 9

What is the size of the angle PRQ if R lies on the circle and PQ is the diameter?

What is the circle theorem that fits this?

Answer 9

It is 90* (a right-angle).

The angle in a semicircle is always a right angle.

Question 10

How do you find the centre of a circle when given 3 points that lie on the circle?

Answer 10

Find the equations of the perpendicular bisectors is the two different chords.

Then find the coordinates of the intersections of the two perpendicular bisectors.