Step by step
The question
Question: write the equation of the circle with center (4, 2) and radius 3.
1 Draw the circle
Draw the grid.
Put the center on the grid: (4, 2).
The radius is 3: go 3 from the center.
Every point 3 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 3 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 = 3².
3² means 3 times 3.
That is 9.
So the equation is (x − 4)² + (y − 2)² = 9.
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: (7, 2).
Put x = 7 and y = 2 in.
We get 3² + 0².
9 plus 0 is 9, 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.