Question

Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options.

The radius of the circle is 3 units.
The center of the circle lies on the x-axis.
The center of the circle lies on the y-axis.
The standard form of the equation is (x – 1)² + y² = 3.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Answer 1

Answers: Choice A, Choice B, Choice E

• A. The radius of the circle is 3 units.
• B. The center of the circle lies on the x-axis.
• E. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

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Step-by-step explanation

Let's complete the square for the x terms.

`\text{x}^2 + \text{y}^2 - 2\text{x} - 8 = 0\\\\(\text{x}^2 - 2\text{x}) + \text{y}^2 - 8 = 0\\\\(\text{x}^2 - 2\text{x}+1-1) + \text{y}^2 - 8 = 0 \ \text{ ... see note below}\\\\(\text{x}^2 - 2\text{x}+1)-1 + \text{y}^2 - 8 = 0\\\\(\text{x}-1)^2 + \text{y}^2 - 9 = 0\\\\(\text{x}-1)^2 + \text{y}^2 = 9\\\\`

Note: The "1" is the result of taking half of the x coefficient (-2) and then squaring the result. This is used to complete the square.

The result
`(\text{x}-1)^2 + \text{y}^2 = 9` is of the form
`(\text{x}-\text{h})^2 + (\text{y}-\text{k})^2 = \text{r}^2` where:

• (h,k) = (1,0) is the center
• r^2 = 9 leads to r = 3 which is the radius.

Since r = 3, it means choice A and choice E are true. Note that the radius of x² + y² = 9 is also r = 3.

Because the y coordinate of the center is y = 0, it means this circle's center is on the horizontal x axis. This makes choice B true.

I used GeoGebra to confirm each answer.