How it works
A circle is the set of all points that are a fixed distance (the radius) from a central point (the center).
The standard form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
- (h, k) is the center of the circle.
- r is the radius.
To write the equation:
- Identify the center (h, k).
- Identify the radius r.
- Substitute h, k, and r into the formula.
- Simplify if needed (like squaring the radius).
Worked example
Write the equation of a circle with center (3, -5) and radius 4.
- 1
Identify the center (h, k).
The center is (3, -5), so h = 3 and k = -5.
- 2
Identify the radius r.
The radius is 4.
- 3
Substitute h, k, and r into the formula: (x - h)^2 + (y - k)^2 = r^2.
Plug in h = 3, k = -5, and r = 4: (x - 3)^2 + (y - (-5))^2 = 4^2.
- 4
Simplify the equation.
(y - (-5)) is the same as (y + 5). 4^2 is 16. So the equation is (x - 3)^2 + (y + 5)^2 = 16.
Answer
(x - 3)^2 + (y + 5)^2 = 16
Common mistakes
- 01
Switching the signs for h and k in the formula.
Why: The formula subtracts h and k, so students sometimes forget to use the opposite sign.
Avoid it: Always write (x - h) and (y - k), and substitute with the actual values, including their signs.
- 02
Forgetting to square the radius.
Why: Students sometimes use 'r' instead of 'r^2' on the right side.
Avoid it: Remember the formula is r squared, so always write r^2 after substituting the radius.
- 03
Incorrect substitution when the center has negative values.
Why: Students may write (y - -5) instead of (y + 5).
Avoid it: Carefully substitute negative values and simplify double negatives.
Practice
Answers stay hidden
1.Write the equation of a circle with center (1, 3) and radius 5.
Show hint
Substitute h = 1, k = 3, r = 5 into the formula.
Reveal answer
(x - 1)^2 + (y - 3)^2 = 25
2.Write the equation of a circle with center (-2, -7) and radius 3.
Show hint
Be careful with the negatives when substituting h and k.
Reveal answer
(x + 2)^2 + (y + 7)^2 = 9
3.Write the equation of a circle with center (0, -4) and radius 8.
Show hint
Remember (x - 0) is just x.
Reveal answer
x^2 + (y + 4)^2 = 64