Question

what is the center of the circle with the equation (x + 5)^2 + (y - 6)^2 = 36?
a. (5,6)
b. (-5,6)
c. (5,-6)
d. (-5,-6)
what is the center of the circle with the equation (x - 3)^2 + (y + 2)^2 = 25?
a. (-3,-2)
b. (3,2)
c. (3,-2)
d. (-3,2)

Explanation:

Step1: Recall the standard form of a circle equation

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: Rewrite the given equation \((x + 5)^2+(y - 6)^2=36\)

We can rewrite \((x + 5)^2\) as \((x-(- 5))^2\). Comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), we have \(h=-5\) and \(k = 6\). So the center is \((-5,6)\) (option b for the first circle equation).

Step3: Rewrite the equation \((x - 3)^2+(y + 2)^2=25\)

We rewrite \((y + 2)^2\) as \((y-(-2))^2\). Comparing with \((x - h)^2+(y - k)^2=r^2\), we get \(h = 3\) and \(k=-2\). So the center is \((3,-2)\) (option c for the second circle equation).

Answer:

For the equation \((x + 5)^2+(y - 6)^2=36\), the answer is b. \((-5,6)\). For the equation \((x - 3)^2+(y + 2)^2=25\), the answer is c. \((3,-2)\).