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

Answer 1

To derive the equation of the circle in standard form given the center \((−1,3)\) and radius \(5\), we can use the standard form of a circle's equation, which is:

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

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

In this case:

- The center \((h, k) = (−1, 3)\)
- The radius \(r = 5\)

Now substituting these values into the equation:

1. Calculate \(r^2\):
\[
r^2 = 5^2 = 25
\]

2. Substitute \(h\), \(k\), and \(r^2\) into the standard equation:
\[
(x - (−1))^2 + (y - 3)^2 = 25
\]
This simplifies to:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]

Therefore, the correct equation of the circle in standard form is:

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

So the correct response is:

**(x + 1)² + (y - 3)² = 25**