Question

1. what is the center of a circle represented by (x - 4)^2+(y + 7)^2 = 81?

a. (-4,7)
b. (4,7)
c. (-4,-7)
d. (4,-7)

1. what is the radius of the circle with the equation x^2 + y^2 = 49?

a. 49
b. 9
c. 14
d. 7

1. what is the standard - form equation of a circle with center at (3, - 2) and radius 4?

a. (x - 3)^2+(y + 2)^2 = 8
b. (x + 3)^2+(y + 2)^2 = 8
c. (x - 3)^2+(y + 2)^2 = 16
d. (x + 3)^2+(y - 2)^2 = 16

Explanation:

Step1: Recall circle - center formula

The standard form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle. For the equation $(x - 4)^2+(y + 7)^2=81$, we have $h = 4$ and $k=-7$. So the center is $(4,-7)$.

Step2: Recall circle - radius formula

The standard - form of a circle equation is $x^{2}+y^{2}=r^{2}$, where $r$ is the radius. For the equation $x^{2}+y^{2}=49$, since $r^{2}=49$ and $r>0$, then $r = 7$.

Step3: Write the standard - form of a circle equation

The standard form of a circle equation with center $(h,k)$ and radius $r$ is $(x - h)^2+(y - k)^2=r^2$. Given center $(3,-2)$ and radius $r = 4$, substituting $h = 3$, $k=-2$ and $r = 4$ into the formula, we get $(x - 3)^2+(y+2)^2=16$.

Answer:

1. d. $(4,-7)$
2. d. $7$
3. c. $(x - 3)^2+(y + 2)^2=16$