Answer: A: (x + 4)² + (y - 5)² = 8
Step-by-step explanation:
The equation of a circle is: (x - h)² + (y - k)² = r² where
- (h, k) is the center of the circle
- r is the radius of the circle
To find the equation of circle A, we need (h, k) and r².
It is given that (h, k) = (-4, 5)
The distance between the center of circles A and B is the radius of circle A.
Let's find the center of circle B by completing the square:
x² - 4x + ____ + y² + 6y + _____ = 12 + ____ + _____
↓ ↓
h = -4/2 k = 6/2
h = -2 k = 3 → Circle B has center (-2, 3)
Now let's find the distance between the center of circles A and B:
[tex](r_A)^2=d^2=(x_A-x_B)^2+(y_A-y_B)^2[/tex]
= (-4 - (-2))² + (5 - 3)²
= (-2)² + 2²
= 4 + 4
= 8
Now we have the the center (h, k) and the r² for circle A:
(h, k) = (-4, 5) and r² = 8 → (x + 4)² + (y - 5)² = 8
