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)

Question

Responses

(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
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+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

Answer 1

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula:

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

In this case, the center of the circle is at \((-1, 3)\) which means \(h = -1\) and \(k = 3\). The radius of the circle is 5, so \(r = 5\).

To derive the equation:

1. Substitute the values of \(h\), \(k\), and \(r\) into the standard form.

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

2. This simplifies to:

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

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

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

So the correct response from the provided options is:

\[ \text{(x+1)² + (y-3)² = 25} \]