Question

which equation represents a circle with center t(5,-1) and a radius of 16 units?
a. ((x - 5)^2 + (y + 1)^2 = 16)
b. ((x - 5)^2 + (y + 1)^2 = 256)
c. ((x + 5)^2 + (y - 1)^2 = 16)
d. ((x + 5)^2 + (y - 1)^2 = 256)

Explanation:

Step1: Recall circle equation formula

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

Step2: Substitute center coordinates

Here, $h=5$, $k=-1$. Substitute into the formula: $(x-5)^2+(y-(-1))^2=r^2$, which simplifies to $(x-5)^2+(y+1)^2=r^2$.

Step3: Calculate squared radius

Radius $r=16$, so $r^2=16^2=256$.

Step4: Final equation

Substitute $r^2=256$ into the equation: $(x-5)^2+(y+1)^2=256$.

Answer:

B. $(x-5)^2+(y+1)^2=256$