Center and radius of x^2 + y^2 - 2y = 15 Step by Step

Step by step

The question

Question: find the center and the radius of x² + y² − 2y = 15.

x² + y² − 2y = 15

1 Group x and y

Put the x terms together in a bracket, and the y terms together.

x² + (y² − 2y) = 15

2 Complete the square for y

To make y² − 2y a full square: half of −2 is −1.

(−1) times (−1) is 1.

So we add 1.

We do it on both sides, so they stay equal.

x² + (y² − 2y + 1) = 15 + 1

−2 ÷ 2 = −1, (−1) × (−1) = 1

Careful!

What we add on the left, we add on the right too.

Then the two sides stay equal.

Careful: add 1 on both sides

3 Write as squares

Each bracket is now a full square.

On the right, 15 + 1 is 16.

So x² + (y − 1)² = 16.

x² + (y − 1)² = 16

y² − 2y + 1 is (y − 1)².

y² − 2y + 1 = (y − 1)²

4 Read the center and radius

Compare with the circle rule: (x − h)² + (y − k)² = r².

(x − h)² + (y − k)² = r²

The signs turn: x² gives h = 0.

y − 1 gives k = 1.

y − 1 → k = 1

x → h = 0

So the center is (0, 1).

center = (0, 1)

r² is 16.

So r is the square root of 16: 4, because 4 times 4 is 16.

r = √16 = 4

So the center is (0, 1) and the radius is 4.

center (0, 1) and r = 4

5 Check

Let's check with a point on the circle: (4, 1), 4 to the right from the center.

Put it in the first equation.

4² + 1² − 2 × 1 = 16 + 1 − 2 = 15

We get 15, the same as the right side.

So the center and the radius are right!

Let's look back.

First: group x and y.

Then: complete the square for y.

Then: write as squares.

Then: read the center and radius.

Last: check.

Answer