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.

(X+1)^2 +(y-3)^2=25
(X-1)^2 +(y-3)^2=25
(X-3)^2+(y-1)^2=25
(X+3)^2+(y-1)^2=25

Answer 1

To derive the equation of a circle in standard form, we use the formula:

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

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

Given:

- Center \((h, k) = (-1, 3)\)
- Radius \(r = 5\)

We can substitute these values into the formula:

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

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

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

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

From the options you provided, the correct equation is:

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