Step by step
The question
Question: write the equation of the circle with center (3, 1) and radius 6.
1 Draw the circle
Draw the grid.
Put the center on the grid: (3, 1).
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 = 1 to y is y − 1.
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 − 1)² = 6².
6² means 6 times 6.
That is 36.
So the equation is (x − 3)² + (y − 1)² = 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: (9, 1).
Put x = 9 and y = 1 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.