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.(1 point)
Responses

(x+3)2+(y−1)2=25
left parenthesis x plus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25

(x−3)2+(y−1)2=25
left parenthesis x minus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25

(x−1)2+(y−3)2=25
left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25

(x+1)2+(y−3)2=25

Answer 1

To derive the equation of a circle in standard form, we use the standard form of a circle's equation, which is given by:

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

where \((h, k)\) is the center of the circle, and \(r\) is the radius.

Given:
- Center: \((-1, 3)\)
- Radius: \(5\)

We substitute the values into the standard form:

- \(h = -1\)
- \(k = 3\)
- \(r = 5\)

Now, plugging in these values:

\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
\[
(x + 1)^2 + (y - 3)^2 = 25
\]

Thus, the equation of the circle in standard form is:

\[
(x + 1)^2 + (y - 3)^2 = 25
\]

Therefore, the correct response from the options is:

\[
\text{(x + 1)}^2 + \text{(y - 3)}^2 = 25
\]