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

Answer 1

To derive the equation of the circle with center at \((-1, 3)\) and radius \(5\), we can use the standard form of the equation of a circle:

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

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

In this case:
- The center \((h, k)\) is \((-1, 3)\), so \(h = -1\) and \(k = 3\).
- The radius \(r\) is \(5\).

Substituting these values into the standard form, we get:

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

This simplifies to:

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

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

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