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

Answer 1

To derive the equation of a circle in standard form, we use the standard form equation of a circle, 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 \((h, k) = (-1, 3)\)
• Radius \(r = 5\)

Substituting these values into the standard form equation:

1. Substitute \(h = -1\), \(k = 3\), and \(r = 5\): \[ (x - (-1))^2 + (y - 3)^2 = 5^2 \] \[ (x + 1)^2 + (y - 3)^2 = 25 \]

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

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

From the given responses, the correct choice is: \[ (x + 1)^2 + (y - 3)^2 = 25 \]