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For a circle defined by the given equation, what are the coordinates of the center and the length of the radius?

[tex]\[ x^2 + y^2 - 4x - 10y + 20 = 0 \][/tex]

A. Center: [tex]\((-2, -5)\)[/tex], Radius: 3 units
B. Center: [tex]\((-2, -5)\)[/tex], Radius: 9 units
C. Center: [tex]\((2, 5)\)[/tex], Radius: 3 units
D. Center: [tex]\((2, 5)\)[/tex], Radius: 9 units

Sagot :

GinnyAnswer
To determine the coordinates of the center and the radius of the circle from the given equation:

[tex]\[ x^2 + y^2 - 4x - 10y + 20 = 0 \][/tex]

we need to put the equation into the standard form of a circle, which is:

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.

1. Rearrange the Equation:

We start with the given equation:

[tex]\[ x^2 + y^2 - 4x - 10y + 20 = 0 \][/tex]

2. Complete the Square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] Terms:

To complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms, we separate and rearrange the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:

[tex]\[ x^2 - 4x + y^2 - 10y = -20 \][/tex]

For [tex]\(x\)[/tex] terms:
The expression [tex]\(x^2 - 4x\)[/tex] can be rewritten by completing the square.
[tex]\[ x^2 - 4x \quad \text{becomes} \quad (x - 2)^2 - 4 \][/tex]

For [tex]\(y\)[/tex] terms:
The expression [tex]\(y^2 - 10y\)[/tex] can be rewritten by completing the square.
[tex]\[ y^2 - 10y \quad \text{becomes} \quad (y - 5)^2 - 25 \][/tex]

3. Rewrite the Equation:

Substitute the completed squares back into the equation:
[tex]\[ (x - 2)^2 - 4 + (y - 5)^2 - 25 = -20 \][/tex]

Combine like terms:
[tex]\[ (x - 2)^2 + (y - 5)^2 - 29 = -20 \][/tex]

Simplify the equation:
[tex]\[ (x - 2)^2 + (y - 5)^2 - 29 = -20 \][/tex]
[tex]\[ (x - 2)^2 + (y - 5)^2 = 9 \][/tex]

4. Identify the Center and Radius:

The equation [tex]\((x - 2)^2 + (y - 5)^2 = 9\)[/tex] matches the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 5\)[/tex]
- [tex]\(r^2 = 9\)[/tex], so [tex]\(r = \sqrt{9} = 3\)[/tex]

Therefore, the coordinates of the center of the circle are [tex]\((2, 5)\)[/tex] and the radius is [tex]\(3\)[/tex] units.

Thus, the correct answer is:
C. center: [tex]\((2, 5)\)[/tex]
radius: 3 units
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