Equations of Circles — Practice
A circle is every point a fixed distance from a fixed point. Write that with the distance formula and you have its equation.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Distance formula — d = √(x2 - x1)2 + (y2 - y1)2.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Completing the square — Rewriting x2 + bx as a perfect square minus a constant.
- Simplest radical form — No perfect square factors left under the root.
Problem 1
Give the centre of (x-3)2 + (y-7)2 = 16.
Worked answer
(3, 7)
Centre (3, 7)
Problem 2
Give the radius of (x-3)2 + (y-7)2 = 16.
Worked answer
4
r2 = 16
r = 4
Problem 3
Give the centre of (x+2)2 + (y-5)2 = 49.
Worked answer
(-2, 5)
x + 2 = x - (-2), so h = -2
Centre (-2, 5)
Problem 4
Write the equation of the circle with centre (0,0) and radius 6.
Worked answer
x^2 + y^2 = 36
x2 + y2 = 36
Problem 5
Write the equation with centre (4, -1) and radius 3.
Worked answer
(x-4)^2 + (y+1)^2 = 9
(x-4)2 + (y+1)2 = 9
Problem 6
Give the radius of (x-1)2 + (y+6)2 = 50 in simplest radical form.
Worked answer
5sqrt(2)
√50 = √25 · 2
= 5√2
Problem 7
A circle has a diameter from (1,2) to (7,10). Find its centre.
Worked answer
(4, 6)
(1+72, 2+102) = (4, 6)
Problem 8
For that same circle, find the radius.
Worked answer
5
Distance = √36+64 = 10
r = 5
Problem 9
Is (6, 8) inside, on, or outside x2 + y2 = 100?
Worked answer
on
36 + 64 = 100
Equal to r2, so it is on the circle.
Problem 10
Is (3, 4) inside, on, or outside x2 + y2 = 100?
Worked answer
inside
9 + 16 = 25
25 < 100, so inside.
Problem 11
A circle has centre (2, -3) and passes through (6, 0). Write its equation.
Worked answer
(x-2)^2 + (y+3)^2 = 25
r = √16 + 9 = 5
(x-2)2 + (y+3)2 = 25
Problem 12
In (x-h)2 + (y-k)2 = r2, the number on the right is
- r2
- r
- 2r
- the diameter
Worked answer
r2
It stores r2.
A right side of 25 means radius 5.
Problem 13
Find the centre of x2 + y2 - 6x + 8y - 11 = 0.
Worked answer
(3, -4)
(x2-6x) + (y2+8y) = 11
Add 9 and 16 to both sides
(x-3)2 + (y+4)2 = 36
Centre (3,-4)
Problem 14
Find the radius of x2 + y2 - 6x + 8y - 11 = 0.
Worked answer
6
(x-3)2 + (y+4)2 = 36
r = 6
Problem 15
Find the radius of x2 + y2 + 10x - 4y + 13 = 0.
Worked answer
4
(x2+10x+25) + (y2-4y+4) = -13+25+4
(x+5)2 + (y-2)2 = 16
r = 4
Problem 16
How many points does the line y = 8 share with x2 + y2 = 25?
Worked answer
0
Radius 5, centred at the origin.
y = 8 is above the top of the circle.
No intersection.