Step by step
The question
Question: write the equation of the circle with center (2, −3) and radius 4.
1 Draw the circle
Draw the grid.
Put the center on the grid: (2, −3).
The radius is 4: go 4 from the center.
Every point 4 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = 2 to x is x − 2.
Go up: from y = −3 to y is y − (−3).
Across and up make a right angle.
The radius 4 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 − 2)² + (y − (−3))² = 4².
Taking away a negative is the same as adding.
So it becomes (x − 2)² + (y + 3)² = 4².
4² means 4 times 4.
That is 16.
So the equation is (x − 2)² + (y + 3)² = 16.
Careful: the signs turn.
Center 2 gives x − 2.
This works for every circle with center (h, k) and radius r.
4 Check
Let's check with a point on the circle: (6, −3).
Put x = 6 and y = −3 in.
We get 4² + 0².
16 plus 0 is 16, 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.