Let x² + y² + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x² at (2, 4). Then A + C is equal to______

Complete Solution:

Since (0, 6) lies on the circle, we substitute it into the equation x² + y² + Ax + By + C = 0:

6B + C = -36     (Equation 1)

Since (2, 4) lies on the circle, substitute it into the circle’s equation:

2A + 4B + C = -20     (Equation 2)

From the tangency condition at (2, 4), we derive:

A + 4B = -36     (Equation 3)

From Equation (3):

A = -36 - 4B

Substitute A = -36 - 4B into Equation (2):

-72 - 8B + 4B + C = -20

Simplify to find C:

C = 52 + 4B

From Equation (1):

6B + (52 + 4B) = -36

Combine terms to find:

B = -8.8

Calculate A:

A = -36 - 4(-8.8) = -0.8

Calculate C:

C = 52 + 4(-8.8) = 16.8

Final Answer:

A = -0.8 and C = 16.8, so A + C = 16.