\(\boxed{ \ x^2 + y^2 + 6x - 24y + 128 = 0 \ }\)
Further explanation
We solve this problem implementing the principle of the general form of the circle equation.
The general form of the circle equation is
\(\boxed{ \ \boxed{ \ x^2 + y^2 + Cx + Dy + E = 0 \ } \ }\)
Given that
- center of circle A(-3, 12)
- the radius is r = 5
Option A
The standard form of the circle equation, in other words the center-radius form, is
\(\boxed{ \ \boxed{ \ (x - h)^2 + (y - k)^2 = r^2 \ } \ }\)
- The center is at (h, k), in this case A(-3, 12)
- The radius represent r = 5
\(\boxed{ \ (x - (-3))^2 + (y - 12)^2 = 5^2 \ }\)
So, we get the standard form of the circle equation.
\(\boxed{ \ (x + 3)^2 + (y - 12)^2 = 25 \ }\)
Expand the equation, implement the square of the binomial pattern.
\(\boxed{ \boxed{ \ (a + b)^2 = a^2 +2ab + b^2 \ } }\)
\(\boxed{ \ x^2 + 6x + 9 + y^2 - 24y + 144 = 25 \ }\)
Rearrange the equation.
\(\boxed{ \ x^2 + y^2 + 6x - 24y + 144 + 9 - 25 = 0 \ }\)
We get the general form of the equation of the given circle.
\(\boxed{ \ \boxed{ \ x^2 + 6x + 9 + y^2 - 24y + 128= 0 \ } \ }\)
Option B
The relationship between the general form and the standard form.
\(\boxed{ \ x^2 + y^2 + Cx + Dy + E = 0 \ } \ \boxed{ \ (x - h)^2 + (y - k)^2 = r^2 \ }\)
\(\boxed{C = -2h} \ \boxed{D = -2k} \ \boxed{E = h^2 + k^2 - r^2}\)
A(-3, 12) → (h, k) and r = 5
\(\boxed{ \ C = -2(-3) = 6 \ } \ \boxed{ \ D = -2(12) = -24 \ } \\\boxed{ \ E = (-3)^2 + 12^2 - 5^2 = 128 \ }\)
Once more, we get the general form of the equation of the given circle.
\(\boxed{ \ \boxed{ \ x^2 + 6x + 9 + y^2 - 24y + 128= 0 \ } \ }\)
Learn more
- Considering the equation of the circle in standard form with center and radius given https://brainly.com/question/5036073
- Investigating similarity between two circles https://brainly.com/question/9177979
- Determining the position of the point against the circle. https://brainly.com/question/4185664
Keywords: circle, general form, equation, center, the radius, explicit, squared binomial, completing square, implicit, center-radius form, conversion
