Center and radius of x^2 + y^2 + 2x + 4y = 4 Step by Step

Step by step

The question

Question: find the center and the radius of x² + y² + 2x + 4y = 4.

x² + y² + 2x + 4y = 4

1 Group x and y

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

(x² + 2x) + (y² + 4y) = 4

2 Complete the square for x

To make x² + 2x 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 + 2x + 1) + (y² + 4y) = 4 + 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 Complete the square for y

To make y² + 4y a full square: half of 4 is 2.

2 times 2 is 4.

So we add 4.

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

(x² + 2x + 1) + (y² + 4y + 4) = 4 + 1 + 4

4 ÷ 2 = 2, 2 × 2 = 4

4 Write as squares

Each bracket is now a full square.

On the right, 4 + 1 + 4 is 9.

So (x + 1)² + (y + 2)² = 9.

(x + 1)² + (y + 2)² = 9

x² + 2x + 1 is (x + 1)².

y² + 4y + 4 is (y + 2)².

x² + 2x + 1 = (x + 1)²

y² + 4y + 4 = (y + 2)²

5 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: x + 1 gives h = −1.

y + 2 gives k = −2.

x + 1 → h = −1

y + 2 → k = −2

So the center is (−1, −2).

center = (−1, −2)

r² is 9.

So r is the square root of 9: 3, because 3 times 3 is 9.

r = √9 = 3

So the center is (−1, −2) and the radius is 3.

center (−1, −2) and r = 3

6 Check

Let's check with a point on the circle: (2, −2), 3 to the right from the center.

Put it in the first equation.

2² + (−2)² + 2 × 2 + 4 × (−2)

= 4 + 4 + 4 − 8 = 4

We get 4, 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 x.

Then: complete the square for y.

Then: write as squares.

Then: read the center and radius.

Last: check.

Answer