Equations of Circles — Drills
Equations of Circles has 7 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Reading the graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Give the radius of (x - 7)2 + (y - 4)2 = 16.
Worked answer
4
r2 = 16
r = 4
Problem 2
Give the centre of (x + 7)2 + (y - 1)2 = 100.
Worked answer
(-7, 1)
h = -7, k = 1
Centre (-7, 1)
Writing the expression
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Write the equation of the circle with centre (1, 3) and radius 11.
Worked answer
(x - 1)^2 + (y - 3)^2 = 121
Substitute h = 1, k = 3, r = 11.
(x - 1)2 + (y - 3)2 = 121
Problem 2
Write the equation of the circle with centre (5, 0) and radius 6.
Worked answer
(x - 5)^2 + (y - 0)^2 = 36
Substitute h = 5, k = 0, r = 6.
(x - 5)2 + (y - 0)2 = 36
Diameter
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A circle has a diameter from (1, 6) to (19, 30). Find its centre.
Worked answer
(10, 18)
(1+192, 6+302) = (10, 18)
Problem 2
A circle has a diameter from (-1, -1) to (29, 39). Find its centre.
Worked answer
(14, 19)
(-1+292, -1+392) = (14, 19)
Point
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is (0, 2) inside, on, or outside x2 + y2 = 64?
Worked answer
inside
02 + 22 = 4
4 < 64
So it is inside.
Problem 2
Is (7, 6) inside, on, or outside x2 + y2 = 100?
Worked answer
inside
72 + 62 = 85
85 < 100
So it is inside.
Form
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In (x-h)2 + (y-k)2 = r2, the number on the right is
- r
- r2
- 2r
- the diameter
Worked answer
r2
It stores r2, not r.
A right side of 25 means a radius of 5.
Completing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the centre of x2 + y2 + 0x - 2y - 80 = 0.
Worked answer
(0, 1)
Group: (x2 + 0x) + (y2 - 2y) = 80
Add 0 and 1 to both sides.
(x - 0)2 + (y - 1)2 = 81
Centre (0, 1)
Problem 2
Find the centre of x2 + y2 - 8x + 2y - 8 = 0.
Worked answer
(4, -1)
Group: (x2 - 8x) + (y2 + 2y) = 8
Add 16 and 1 to both sides.
(x - 4)2 + (y + 1)2 = 25
Centre (4, -1)
Intersect
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many points does the line y = 7 share with x2 + y2 = 16?
Worked answer
0
Radius 4, centred at the origin, so it reaches y = 4.
y = 7 is above that, so they never meet.
Problem 2
How many points does the line y = 7 share with x2 + y2 = 25?
Worked answer
0
Radius 5, centred at the origin, so it reaches y = 5.
y = 7 is above that, so they never meet.