A circle is drawn on a coordinate plane with the center 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

Answer 1

To derive the equation of the circle, we can use the standard form of the equation of a circle, which is:

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

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

Given that the center of the circle is (-1,3) and the radius is 5, we can plug these values into the standard form equation:

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

Simplify:

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

Expanding:

x^2 + 2x + 1 + y^2 - 6y + 9 = 25

Rearrange and combine like terms:

x^2 + y^2 + 2x - 6y + 10 = 25

Subtract 25 from both sides:

x^2 + y^2 + 2x - 6y - 15 = 0

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

x^2 + y^2 + 2x - 6y - 15 = 0