Step by step
The question
Question: write the equation of the circle with center (−4, −2) and radius 5.
1 Draw the circle
Draw the grid.
Put the center on the grid: (−4, −2).
The radius is 5: go 5 from the center.
Every point 5 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = −4 to x is x − (−4).
Go up: from y = −2 to y is y − (−2).
Across and up make a right angle.
The radius 5 is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
Put in a, b and c: (x − (−4))² + (y − (−2))² = 5².
Taking away a negative is the same as adding.
So it becomes (x + 4)² + (y + 2)² = 5².
5² means 5 times 5.
That is 25.
So the equation is (x + 4)² + (y + 2)² = 25.
Careful: the signs turn.
Center −4 gives x + 4.
This works for every circle with center (h, k) and radius r.
4 Check
Let's check with a point on the circle: (1, −2).
Put x = 1 and y = −2 in.
We get 5² + 0².
25 plus 0 is 25, the number on the right.
Right!
Let's look back.
First: draw the circle.
Then: across and up.
Then: use a² + b² = c².
Last: check.