Question

select the correct answer.
circle c has a center at (-2,10) and contains the point p(10,5). which equation represents circle c?
a. ((x - 2)^2 + (y + 10)^2 = 13)
b. ((x - 2)^2 + (y + 10)^2 = 169)
c. ((x + 2)^2 + (y - 10)^2 = 13)
d. ((x + 2)^2 + (y - 10)^2 = 169)

Explanation:

Step1: Recall circle equation form

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

Step2: Plug in center coordinates

Center is $(-2,10)$, so $h=-2$, $k=10$. Substitute: $(x-(-2))^2+(y-10)^2=r^2$ → $(x+2)^2+(y-10)^2=r^2$

Step3: Calculate radius squared

Use point $P(10,5)$: $r^2=(10-(-2))^2+(5-10)^2$
$r^2=(12)^2+(-5)^2=144+25=169$

Step4: Final circle equation

Substitute $r^2=169$: $(x+2)^2+(y-10)^2=169$

Answer:

D. $(x + 2)^2 + (y - 10)^2 = 169$