Question
Dedric is trying to graph a circle on a computer software program that has a grid. The equation he is graphing is x^2 + y^2 - 20x - 14y + 5 = 0. Where would he place the center of the circle and what is its radius?
a. center = (-20, -14) radius = 8
b. center = (20,14) radius = 5
c. center = (-10, -7) radius = 5
d. center = (10,7) radius = 12
a. center = (-20, -14) radius = 8
b. center = (20,14) radius = 5
c. center = (-10, -7) radius = 5
d. center = (10,7) radius = 12
Answers
Answered by
GPT-4o mini
To find the center and radius of the circle represented by the equation \( x^2 + y^2 - 20x - 14y + 5 = 0 \), we first need to rewrite the equation in standard form.
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.
We start by rearranging the given equation:
1. Move the constant to the other side:
\[
x^2 + y^2 - 20x - 14y = -5
\]
2. Now we will complete the square for the \(x\) and \(y\) terms.
**For the \(x\) terms:**
- Take \(x^2 - 20x\).
- To complete the square, take half of the coefficient of \(x\) (which is \(-20\)), square it: \((\frac{-20}{2})^2 = 100\).
- Thus, we rewrite \(x^2 - 20x\) as \((x - 10)^2 - 100\).
**For the \(y\) terms:**
- Take \(y^2 - 14y\).
- Take half of the coefficient of \(y\) (which is \(-14\)), square it: \((\frac{-14}{2})^2 = 49\).
- Thus, we rewrite \(y^2 - 14y\) as \((y - 7)^2 - 49\).
Putting this all together, we have:
\[
(x - 10)^2 - 100 + (y - 7)^2 - 49 = -5
\]
Combine terms:
\[
(x - 10)^2 + (y - 7)^2 - 149 = -5
\]
Rearranging gives:
\[
(x - 10)^2 + (y - 7)^2 = 144
\]
Now we can identify the center and radius:
- The center \((h, k)\) is \((10, 7)\).
- The radius \(r\) is \(\sqrt{144} = 12\).
Thus, the center is \((10, 7)\) and the radius is \(12\).
The correct answer is:
**d. center = (10, 7) radius = 12**.
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.
We start by rearranging the given equation:
1. Move the constant to the other side:
\[
x^2 + y^2 - 20x - 14y = -5
\]
2. Now we will complete the square for the \(x\) and \(y\) terms.
**For the \(x\) terms:**
- Take \(x^2 - 20x\).
- To complete the square, take half of the coefficient of \(x\) (which is \(-20\)), square it: \((\frac{-20}{2})^2 = 100\).
- Thus, we rewrite \(x^2 - 20x\) as \((x - 10)^2 - 100\).
**For the \(y\) terms:**
- Take \(y^2 - 14y\).
- Take half of the coefficient of \(y\) (which is \(-14\)), square it: \((\frac{-14}{2})^2 = 49\).
- Thus, we rewrite \(y^2 - 14y\) as \((y - 7)^2 - 49\).
Putting this all together, we have:
\[
(x - 10)^2 - 100 + (y - 7)^2 - 49 = -5
\]
Combine terms:
\[
(x - 10)^2 + (y - 7)^2 - 149 = -5
\]
Rearranging gives:
\[
(x - 10)^2 + (y - 7)^2 = 144
\]
Now we can identify the center and radius:
- The center \((h, k)\) is \((10, 7)\).
- The radius \(r\) is \(\sqrt{144} = 12\).
Thus, the center is \((10, 7)\) and the radius is \(12\).
The correct answer is:
**d. center = (10, 7) radius = 12**.