Answer :
To determine the coordinates of the center and the radius of the circle from the given equation [tex]\(x^2 + y^2 - 12y - 20.25 = 0\)[/tex], let's proceed with the following steps:
1. Rewrite the Equation:
The given equation is [tex]\(x^2 + y^2 - 12y - 20.25 = 0\)[/tex].
2. Complete the Square:
To rewrite the equation in the standard form of a circle's equation, we need to complete the square for the [tex]\(y\)[/tex]-term.
First, rearrange the equation:
[tex]\[ x^2 + (y^2 - 12y) - 20.25 = 0 \][/tex]
3. Complete the Square for the [tex]\(y\)[/tex]-term:
Consider the [tex]\(y\)[/tex]-term [tex]\(y^2 - 12y\)[/tex]. To complete the square, add and subtract [tex]\((\frac{12}{2})^2 = 36\)[/tex]:
[tex]\[ y^2 - 12y = (y - 6)^2 - 36 \][/tex]
Substitute this back into the equation:
[tex]\[ x^2 + (y - 6)^2 - 36 - 20.25 = 0 \][/tex]
4. Simplify the Equation:
Combine like terms:
[tex]\[ x^2 + (y - 6)^2 - 56.25 = 0 \][/tex]
[tex]\[ x^2 + (y - 6)^2 = 56.25 \][/tex]
5. Identify the Center and Radius:
The equation is now in the standard form of a circle's equation [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. By comparing:
- [tex]\((x - 0)^2 + (y - 6)^2 = 56.25\)[/tex],
We can see that the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((0, 6)\)[/tex] and [tex]\(r^2 = 56.25\)[/tex].
6. Calculate the Radius:
The radius [tex]\(r\)[/tex] is the square root of 56.25:
[tex]\[ r = \sqrt{56.25} = 7.5 \][/tex]
Therefore, the coordinates of the center of the circle are [tex]\((0, 6)\)[/tex] and the radius is 7.5. This matches option (1).
So, the correct answer is:
[tex]\( \boxed{\text{center } (0, 6) \text{ and radius } 7.5} \)[/tex]
1. Rewrite the Equation:
The given equation is [tex]\(x^2 + y^2 - 12y - 20.25 = 0\)[/tex].
2. Complete the Square:
To rewrite the equation in the standard form of a circle's equation, we need to complete the square for the [tex]\(y\)[/tex]-term.
First, rearrange the equation:
[tex]\[ x^2 + (y^2 - 12y) - 20.25 = 0 \][/tex]
3. Complete the Square for the [tex]\(y\)[/tex]-term:
Consider the [tex]\(y\)[/tex]-term [tex]\(y^2 - 12y\)[/tex]. To complete the square, add and subtract [tex]\((\frac{12}{2})^2 = 36\)[/tex]:
[tex]\[ y^2 - 12y = (y - 6)^2 - 36 \][/tex]
Substitute this back into the equation:
[tex]\[ x^2 + (y - 6)^2 - 36 - 20.25 = 0 \][/tex]
4. Simplify the Equation:
Combine like terms:
[tex]\[ x^2 + (y - 6)^2 - 56.25 = 0 \][/tex]
[tex]\[ x^2 + (y - 6)^2 = 56.25 \][/tex]
5. Identify the Center and Radius:
The equation is now in the standard form of a circle's equation [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. By comparing:
- [tex]\((x - 0)^2 + (y - 6)^2 = 56.25\)[/tex],
We can see that the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((0, 6)\)[/tex] and [tex]\(r^2 = 56.25\)[/tex].
6. Calculate the Radius:
The radius [tex]\(r\)[/tex] is the square root of 56.25:
[tex]\[ r = \sqrt{56.25} = 7.5 \][/tex]
Therefore, the coordinates of the center of the circle are [tex]\((0, 6)\)[/tex] and the radius is 7.5. This matches option (1).
So, the correct answer is:
[tex]\( \boxed{\text{center } (0, 6) \text{ and radius } 7.5} \)[/tex]
