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Consider a circle whose equation is [tex]$x^2 + y^2 - 2x - 8 = 0$[/tex]. Which statements are true? Select three options.

A. The radius of the circle is 3 units.
B. The center of the circle lies on the [tex]$x$[/tex]-axis.
C. The center of the circle lies on the [tex]$y$[/tex]-axis.
D. The standard form of the equation is [tex]$(x-1)^2 + y^2 = 9$[/tex].
E. The radius of this circle is the same as the radius of the circle whose equation is [tex]$x^2 + y^2 = 9$[/tex].

Sagot :

GinnyAnswer
Let's analyze the equation of the circle and verify the truth of each statement given.

Step 1: Rewrite the circle equation in standard form

The given circle equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]

To rewrite this equation in standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.

Step 2: Completing the square

Consider the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
Complete the square:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]

Now, consider the terms involving [tex]\(y\)[/tex]:
[tex]\[ y^2 \][/tex]
This term is already a perfect square:
[tex]\[ y^2 = (y - 0)^2 \][/tex]

Substitute these back into the equation:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]

Combine constants on one side:
[tex]\[ (x - 1)^2 + (y^2) = 9 \][/tex]

So, the standard form of the circle's equation is:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]

Step 3: Identifying the center and radius

From the standard form equation:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]

The center [tex]\((h, k)\)[/tex] of the circle is:
[tex]\[ (h, k) = (1, 0) \][/tex]

The radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{9} = 3 \][/tex]

Step 4: Verifying the statements

1. The radius of the circle is 3 units.
- True. We found the radius to be 3 units.

2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center [tex]\((1, 0)\)[/tex] means [tex]\(k = 0\)[/tex], so it lies on the [tex]\(x\)[/tex]-axis.

3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center is [tex]\((1, 0)\)[/tex], so it does not lie on the [tex]\(y\)[/tex]-axis.

4. The standard form of the equation is [tex]\((x - 1)^2 + y^2 = 3\)[/tex].
- False. The correct standard form is [tex]\((x - 1)^2 + y^2 = 9\)[/tex].

5. The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
- True. The radius of the circle with the equation [tex]\(x^2 + y^2 = 9\)[/tex] is 3, which matches our circle's radius.

Conclusion

The true statements are:
- The radius of the circle is 3 units.
- The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
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