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
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