Write the equation in general conic form of the circle that passes through the points (0, 0), (0, 6), and (2, 3).
2x^2 + 2y^2 + x - y = 0

2x² + 5x + 2y² − 12y = 0

Step-by-step explanation:

Equation of a circle is:

(x − h)² + (y − k)² = r²

Given three points, we can write three equations:

(0 − h)² + (0 − k)² = r²

(0 − h)² + (6 − k)² = r²

(2 − h)² + (3 − k)² = r²

Simplifying:

h² + k² = r²

h² + 36 − 12k + k² = r²

4 − 4h + h² + 9 − 6k + k² = r²

Setting the first two equations equal:

h² + k² = h² + 36 − 12k + k²

0 = 36 − 12k

k = 3

Setting the first and third equations equal:

h² + k² = 4 − 4h + h² + 9 − 6k + k²

0 = 4 − 4h + 9 − 6k

4h = 13 − 6k

4h = 13 − 6(3)

4h = -5

h = -5/4

Plugging into the first equation:

h² + k² = r²

(-5/4)² + (3)² = r²

r² = 169/16

r = 13/4

Therefore, the equation of the circle is:

(x + 5/4)² + (y − 3)² = 169/16

Converting to conic form:

x² + 5/2 x + 25/16 + y² − 6y + 9 = 169/16

16x² + 40x + 25 + 16y² − 96y + 144 = 169

16x² + 40x + 16y² − 96y = 0

2x² + 5x + 2y² − 12y = 0

Graph:

desmos.com/calculator/ilzq3xgvbl

1: -2 3/4, -2 1/2, -14, 0, 3, 8 1/3

2: -4, -3, -2 1/2, -2 1/4, 3 1/4,

step-by-step explanation: