Step by step
The question
Question: write the equation of the circle with center (−3, 0) and radius 6.
1 Draw the circle
Draw the grid.
Put the center on the grid: (−3, 0).
The radius is 6: go 6 from the center.
Every point 6 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = −3 to x is x − (−3).
Go up: from y = 0 to y is y − 0.
Across and up make a right angle.
The radius 6 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 − (−3))² + (y − 0)² = 6².
Taking away a negative is the same as adding.
Taking away 0 changes nothing.
So it becomes (x + 3)² + y² = 6².
6² means 6 times 6.
That is 36.
So the equation is (x + 3)² + y² = 36.
Careful: the signs turn.
Center −3 gives x + 3.
This works for every circle with center (h, k) and radius r.
4 Check
Let's check with a point on the circle: (3, 0).
Put x = 3 and y = 0 in.
We get 6² + 0².
36 plus 0 is 36, 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.