Question

the center of the circle is at (-1, 1).
what is the equation of the circle?
a. ((x - 1)^2 + (y + 1)^2 = 8)
b. ((x + 1)^2 + (y - 1)^2 = 8)
c. ((x - 1)^2 + (y + 1)^2 = 16)
d. ((x + 1)^2 + (y - 1)^2 = 16)

Explanation:

Step1: Recall circle equation formula

The standard equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius.

Step2: Plug in center coordinates

Given center $(-1,1)$, so $h=-1$, $k=1$. Substitute into formula: $(x-(-1))^2 + (y-1)^2 = r^2$, which simplifies to $(x+1)^2 + (y-1)^2 = r^2$.

Step3: Calculate radius from graph

Count units from center $(-1,1)$ to a point on the circle, e.g., $(3,1)$. Radius $r = 3 - (-1) = 4$. Then $r^2 = 4^2 = 16$.

Step4: Finalize the equation

Substitute $r^2=16$ into the equation: $(x+1)^2 + (y-1)^2 = 16$.

Answer:

D. $(x + 1)^2 + (y - 1)^2 = 16$