Question

what is the radius of the circle with the equation ((x - 1)^2 + (y - 4)^2 = 49)?
a. 7
b. 4
c. 6
d. 5

which radius corresponds to the circle with the equation ((x + 5)^2 + (y - 1)^2 = 121)?
a. 11
b. 12
c. 121
d. 13

what is the equation of a circle with center ((1, 3)) and radius 5?
a. ((x + 1)^2 + (y + 3)^2 = 25)
b. ((x + 1)^2 + (y + 3)^2 = 5)
c. ((x - 1)^2 + (y - 3)^2 = 25)
d. ((x - 1)^2 + (y - 3)^2 = 5)

what is the center of the circle given by ((x + 3)^2 + (y - 4)^2 = 49)?
a. ((3, -4))
b. ((-3, 4))
c. ((-3, -4))
d. ((3, 4))

Explanation:

Step1: Recall the standard form of a circle's equation

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

Step2: Solve for the first question

For the equation \((x - 1)^2+(y - 4)^2 = 49\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(r^2 = 49\), so \(r=\sqrt{49}=7\).

Step3: Solve for the second question

For the equation \((x + 5)^2+(y - 1)^2 = 121\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(r^2 = 121\), so \(r=\sqrt{121}=11\).

Step4: Solve for the third question

Given center \((1,3)\) and radius \(r = 5\), using the standard form \((x - h)^2+(y - k)^2=r^2\), we get \((x - 1)^2+(y - 3)^2=25\).

Step5: Solve for the fourth question

For the equation \((x + 3)^2+(y - 4)^2 = 49\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h=-3,k = 4\), so the center is \((-3,4)\).

Answer:

1. a. 7
2. a. 11
3. c. \((x - 1)^2+(y - 3)^2=25\)
4. b. \((-3,4)\)