The Equation of a Circle

Mathematics · Geometry · ages 16-17

Name ______________________ Date ____________

A circle is the set of all the points that sit a fixed distance from one chosen point.
Circle one: True False
In (x - a)² + (y - b)² = r², the number on the right-hand side is the radius.
Circle one: True False
A circle has centre (2, 3) and radius 4, so its equation is (x - 2)² + (y - 3)² = k. What is k?
Answer: ______________
What is the radius of the circle (x + 1)² + (y - 4)² = 49?
Answer: ______________
Write the equation of the circle with centre (-4, 7) and radius 11.
- (x - 4)² + (y + 7)² = 121
- (x + 4)² + (y - 7)² = 11
- (x - 4)² + (y - 7)² = 121
- (x + 4)² + (y - 7)² = 121
Leo reads the radius of (x - 1)² + (y + 5)² = 36 straight off as 36. Where is the slip?
- 36 is the radius squared, so he needs its square root, which is 6
- the radius is read off the brackets, not the right-hand side
- the radius is half of 36, so it is 18
- no radius can be found until the centre is known
Where does the point (1, 7) sit relative to the circle (x - 1)² + (y - 3)² = 25?
- on the circle
- inside the circle
- outside the circle
- on the circle, because the point shares the centre's x-coordinate
Mia writes the circle with centre (5, -2) and radius 3 as (x + 5)² + (y - 2)² = 9. Where is the slip?
- the radius should have been squared before it was written down
- the centre and the radius were written in the wrong order
- both bracket signs are flipped, so it is (x - 5)² + (y + 2)² = 9
- a negative coordinate cannot appear in a circle equation
Which of these points lies inside the circle x² + y² = 26?
- (5, 5)
- (1, 5)
- (0, 6)
- (3, 4)
A circle has centre (5, 2) and passes through the point (13, 8). What is its equation?
- (x - 5)² + (y - 2)² = 10
- (x + 5)² + (y + 2)² = 100
- (x - 5)² + (y - 2)² = 100
- (x - 5)² + (y - 2)² = 14

Answer key

For grown-ups. Fold this page away before handing over the rest.

The Equation of a Circle W1-mt_YARqmi_ioe-s1

True · That is exactly what a circle is, and it is where the equation comes from. The chosen point is the centre and the fixed distance is the radius.
False · The right-hand side is the radius squared. The equation came from squaring the distance, so the radius got squared with it.
16 · The right-hand side is the radius squared, so k = 4² = 16.
7 · The right-hand side is r², so r² = 49 and the radius is 7.
(x + 4)² + (y - 7)² = 121 · The centre (-4, 7) flips to (x + 4)² + (y - 7)², and the radius 11 goes on the right squared, so the right-hand side is 121.
36 is the radius squared, so he needs its square root, which is 6 · The 36 is the radius squared, because the equation came from squaring the distance. The radius is the square root of 36, which is 6.
inside the circle · Substituting gives (1 - 1)² + (7 - 3)² = 0 + 16 = 16, which is smaller than 25, so the point is inside.
both bracket signs are flipped, so it is (x - 5)² + (y + 2)² = 9 · Each bracket carries the opposite sign of its coordinate. A centre of (5, -2) gives (x - 5)² + (y + 2)². Mia did square the radius correctly.
(3, 4) · Inside means x² + y² is below 26. The four values are 50, 26, 36 and 25, so only (3, 4) is inside. (1, 5) gives exactly 26, so it is on the circle.
(x - 5)² + (y - 2)² = 100 · The radius is the distance from the centre to that point, so r² = (13 - 5)² + (8 - 2)² = 64 + 36 = 100. The equation is (x - 5)² + (y - 2)² = 100.