Pre-calculus question?

Question

Find the standard form of the equation of a circle with solutions at A(0.0), B(0,5), and C(3,3)

Answer 1

The standard form of circle equation (x - h)² + (y - k)² = r², where r = radius (h, k) = center.

The solutions of the circle equation are A(0.0), B(0,5) and C(3,3).

Substitute (x, y) = (0, 0) in the standard form of circle equation.

(0 - h)² + (0 - k)² = r² ⇒ h² + k² = r² ------> equation 1.

Substitute (x, y) = (0, 5) in the standard form of circle equation.

(0 - h)² + (5 - k)² = r²

h² + 25 - 10k + k² = r²

h² + k² + 25 - 10k = r²

Substitute h² + k² = r² in the above equation.

r² + 25 - 10k = r²

10k = 25

k = 25/10 = 5/2 = 2.5

 

Substitute (x, y) = (3, 3) in the standard form of circle equation.

(3 - h)² + (3 - k)² = r²

9 + - 6h + h² + 9 - 6k + k² = r²

h² + k² + 18 - 6h - 6k = r²

Substitute h² + k² = r² and k = 5/2 in the above equation.

r² + 18 - 6h - 6(5/2) = r²

18 - 6h - 15 =0

- 6h = - 3

h = (-3)/(-6) = 1/2 = 0.5.

 

Substitute h = 0.5 and k = 2.5 in the equation r² = h² + k².

r² = (0.5)² + (2.5)²

r² = 0.25 + 6.25

r² = 6.5.

 

The standard form of circle equation (x - 0.5)² + (y - 2.5)² = 6.5.