Question

what is the radius of the circle with the equation (x - 2)^2+(y + 1)^2 = 4?
a. 4
b. 2
c. 1
d. 8
what is the equation of a circle with center at (-4,3) and radius 5?
a. (x - 4)^2+(y + 3)^2 = 125
b. (x - 4)^2+(y + 3)^2 = 5
c. (x + 4)^2+(y - 3)^2 = 25
d. (x + 4)^2+(y - 3)^2 = 5
which of the following is the equation of a circle with center (-5,-5) and radius 10?
a. (x - 5)^2+(y - 5)^2 = 100
b. (x + 5)^2+(y + 5)^2 = 100
c. (x + 5)^2+(y + 5)^2 = 20
d. (x - 5)^2+(y - 5)^2 = 10

Explanation:

Step1: Recall circle - equation formula

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

First question

The equation of the circle is \((x - 2)^2+(y + 1)^2 = 4\). Comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(r^2=4\).

Step2: Solve for \(r\)

Take the square - root of both sides of \(r^2 = 4\). Since \(r>0\), \(r = 2\). The answer is b.

Second question

The center of the circle is \((-4,3)\) and the radius \(r = 5\). Substitute \(h=-4\), \(k = 3\), and \(r = 5\) into the standard form \((x - h)^2+(y - k)^2=r^2\).
We get \((x-(-4))^2+(y - 3)^2=5^2\), which simplifies to \((x + 4)^2+(y - 3)^2=25\). The answer is c.

Third question

The center of the circle is \((-5,-5)\) and the radius \(r = 10\). Substitute \(h=-5\), \(k=-5\), and \(r = 10\) into the standard form \((x - h)^2+(y - k)^2=r^2\).
We get \((x-(-5))^2+(y-(-5))^2=10^2\), which simplifies to \((x + 5)^2+(y + 5)^2=100\). The answer is b.

Answer:

First question: b. 2
Second question: c. \((x + 4)^2+(y - 3)^2=25\)
Third question: b. \((x + 5)^2+(y + 5)^2=100\)