Step by step
The question
Question: find the center and the radius of (x − 1)² + (y + 2)² = 13.
1 Compare with the rule
Every circle has this form: (x − h)² + (y − k)² = r².
The center is (h, k).
The radius is r.
2 Find the center
x − 1 is like x − h.
So h = 1.
y + 2 is the same as y − (−2).
Taking away a negative is adding.
So k = −2.
Careful: the sign turns.
+ 2 in the bracket means the center has −2.
So the center is (1, −2).
3 Find the radius
r² is the number on the right: r² = 13.
r times r is 13.
So r is the square root of 13.
13 is not a square number.
So we keep √13.
That is the exact answer.
With a calculator, √13 is about 3.6, to the nearest tenth.
So the center is (1, −2) and the radius is √13.
4 Check
Let's check: draw the circle.
Center (1, −2), radius about 3.6.
Put the center on the grid.
Draw the circle around it.
This point is on the circle: (3, 1).
It is 2 across and 3 up from the center.
Put x = 3 and y = 1 in.
We get 2² + 3².
4 plus 9 is 13, the number on the right.
Right!
Let's look back.
First: compare with the rule.
Then: find the center.
Then: find the radius.
Last: check.