Question

which of the following equations represents a circle with a center at (-1,3) and radius 2?
a. (x - 1)^2 + (y + 3)^2 = 16
b. (x - 1)^2 + (y + 3)^2 = 4
c. (x + 1)^2 + (y + 3)^2 = 4
d. (x + 1)^2 + (y - 3)^2 = 4
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)

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 and \(r\) is the radius.

Step2: Analyze the first question

For a circle with center \((-1,3)\) and radius \(2\), substitute \(h=-1\), \(k = 3\), \(r = 2\) into the standard form.
We get \((x-(-1))^2+(y - 3)^2=2^2\), which simplifies to \((x + 1)^2+(y - 3)^2=4\).

Step3: Analyze the second question

For the equation \((x - 4)^2+(y+7)^2=81\), compare it with the standard form \((x - h)^2+(y - k)^2 = r^2\).
Here \(h = 4\), \(k=-7\), so the center is \((4,-7)\).

Answer:

First question: d. \((x + 1)^2+(y - 3)^2=4\)
Second question: c. \((4,-7)\)