Question

1. what is the center of the circle with the equation (x + 4)^2+(y - 3)^2 = 18?

a. (-4,-3)
b. (-4,3)
c. (4,-3)
d. (4,3)

1. what is the center of the circle with the equation (x - 4)^2+(y + 5)^2 = 100?

a. (-4,5)
b. (-4,-5)
c. (4,5)
d. (4,-5)

1. which of the following represents a circle with its center at the origin and a radius of 10?

a. x^2 + y^2 = 100
b. x^2 - y^2 = 100
c. x^2 + y^2 = 10
d. (x + 10)^2+(y + 10)^2 = 100

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.

Step2: Find center of first circle

For the equation $(x + 4)^2+(y - 3)^2 = 18$, we have $x-(-4)^2+(y - 3)^2=18$. So the center is $(-4,3)$.

Step3: Find center of second circle

For the equation $(x - 4)^2+(y + 5)^2 = 100$, we have $(x - 4)^2+(y-(-5))^2 = 100$. So the center is $(4,-5)$.

Step4: Check circle - at - origin equation

The equation of a circle with center at the origin $(0,0)$ and radius $r$ is $x^2 + y^2=r^2$. If $r = 10$, then $r^2=100$ and the equation is $x^2 + y^2=100$.

Answer:

1. B. $(-4,3)$
2. D. $(4,-5)$
3. A. $x^2 + y^2=100$