Step by step
The question
Question: write the equation of the circle with center (1, 5) and radius 2.
1 Draw the circle
Draw the grid.
Put the center on the grid: (1, 5).
The radius is 2: go 2 from the center.
Every point 2 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = 1 to x is x − 1.
Go up: from y = 5 to y is y − 5.
Across and up make a right angle.
The radius 2 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 − 1)² + (y − 5)² = 2².
2² means 2 times 2.
That is 4.
So the equation is (x − 1)² + (y − 5)² = 4.
Careful: the signs turn.
Center 1 gives x − 1.
This works for every circle with center (h, k) and radius r.
4 Check
Let's check with a point on the circle: (3, 5).
Put x = 3 and y = 5 in.
We get 2² + 0².
4 plus 0 is 4, 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.