Center and radius of x^2 + y^2 - 2x

Step by step

The question

Question: find the center and the radius of x² + y² − 2x − 4y = 0.

x² + y² − 2x − 4y = 0

1 Group x and y

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

(x² − 2x) + (y² − 4y) = 0

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) = 0 + 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) = 0 + 1 + 4

−4 ÷ 2 = −2, (−2) × (−2) = 4

4 Write as squares

Each bracket is now a full square.

On the right, 0 + 1 + 4 is 5.

So (x − 1)² + (y − 2)² = 5.

(x − 1)² + (y − 2)² = 5

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 5.

5 is not a square number.

So we keep √5.

That is the exact answer.

r = √5

With a calculator, √5 is about 2.2, to the nearest tenth.

√5 ≈ 2.2

So the center is (1, 2) and the radius is √5.

center (1, 2) and r = √5

6 Check

Let's check with a point on the circle: (2, 4), 1 across and 2 up from the center.

Put it in the first equation.

2² + 4² − 2 × 2 − 4 × 4 = 4 + 16 − 4 − 16

= 0

We get 0, 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