Center and radius of x^2 + y^2 - 10x = 0 Step by Step

Step by step

The question

Question: find the center and the radius of x² + y² − 10x = 0.

x² + y² − 10x = 0

1 Group x and y

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

(x² − 10x) + y² = 0

2 Complete the square for x

To make x² − 10x a full square: half of −10 is −5.

(−5) times (−5) is 25.

So we add 25.

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

(x² − 10x + 25) + y² = 0 + 25

−10 ÷ 2 = −5, (−5) × (−5) = 25

Careful!

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

Then the two sides stay equal.

Careful: add 25 on both sides

3 Write as squares

Each bracket is now a full square.

On the right, 0 + 25 is 25.

So (x − 5)² + y² = 25.

(x − 5)² + y² = 25

x² − 10x + 25 is (x − 5)².

x² − 10x + 25 = (x − 5)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: x − 5 gives h = 5.

y² gives k = 0.

x − 5 → h = 5

y → k = 0

So the center is (5, 0).

center = (5, 0)

r² is 25.

So r is the square root of 25: 5, because 5 times 5 is 25.

r = √25 = 5

So the center is (5, 0) and the radius is 5.

center (5, 0) and r = 5

5 Check

Let's check with a point on the circle: (10, 0), 5 to the right from the center.

Put it in the first equation.

10² + 0² − 10 × 10 = 100 + 0 − 100 = 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: write as squares.

Then: read the center and radius.

Last: check.

Answer