Question
a circle is drawn on a coordinate plane with the center point at (-1, 3) and a radius of 5. derive the equation of the circle in standard form, using the given information and the pythagorean theorem.
a. (x + 1)^2 + (y - 3)^2 = 25
b. (x - 3)^2 + (y - 1)^2 = 25
c. (x - 1)^2 + (y - 3)^2 = 25
d. (x + 3)^2 + (y - 1)^2 = 25
a. (x + 1)^2 + (y - 3)^2 = 25
b. (x - 3)^2 + (y - 1)^2 = 25
c. (x - 1)^2 + (y - 3)^2 = 25
d. (x + 3)^2 + (y - 1)^2 = 25
Answers
Answered by
GPT-4o mini
To derive the equation of a circle in standard form, we can use the formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- Center \((-1, 3)\): Here, \(h = -1\) and \(k = 3\).
- Radius \(r = 5\).
Now, we can plug these values into the formula:
1. First, substitute \(h\) and \(k\):
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
This simplifies to:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Therefore, the correct answer is:
**a. (x + 1)^2 + (y - 3)^2 = 25**
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- Center \((-1, 3)\): Here, \(h = -1\) and \(k = 3\).
- Radius \(r = 5\).
Now, we can plug these values into the formula:
1. First, substitute \(h\) and \(k\):
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
This simplifies to:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Therefore, the correct answer is:
**a. (x + 1)^2 + (y - 3)^2 = 25**