Question

what is the equation of a circle with center (-2,1) and radius 6?
a. (x + 2)² + (y - 1)² = 36
b. (x - 2)² + (y + 1)² = 6
c. (x - 2)² + (y + 1)² = 36
d. (x + 2)² + (y + 1)² = 36

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)

what is the center of the circle described by the equation (x - 9)² + (y + 10)² = 81?
a. (-9,-10)
b. (9,-10)
c. (9,10)
d. (-9,10)

what is the radius of the circle described by the equation (x - 8)² + (y + 9)² = 225?
a. 225
b. 13
c. 15
d. 14

what is the center of the circle with the equation (x - 7)² + (y + 8)² = 40?
a. (-7,8)
b. (-7,-8)
c. (7,8)
d. (7,-8)

which of the following is the standard form of a circle with center (0, -1) and radius 4?
a. (x)² + (y - 1)² = 4
b. (x)² + (y - 1)² = 16
c. (x)² + (y + 1)² = 4
d. (x)² + (y + 1)² = 16

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: Solve the first question

Given center \((-2,1)\) and radius \(6\). Here \(h=-2,k = 1,r = 6\). Substitute into the formula: \((x-(-2))^2+(y - 1)^2=6^2\), which simplifies to \((x + 2)^2+(y - 1)^2=36\).

Step3: Solve the second question

For the equation \((x - 3)^2+(y+2)^2=25\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h = 3,k=-2\). So the center is \((3,-2)\).

Step4: Solve the third question

For the equation \((x - 9)^2+(y + 10)^2=81\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h = 9,k=-10\). So the center is \((9,-10)\).

Step5: Solve the fourth question

For the equation \((x - 8)^2+(y + 9)^2=225\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(r^2=225\), so \(r=\sqrt{225}=15\).

Step6: Solve the fifth question

For the equation \((x - 7)^2+(y+8)^2=40\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h = 7,k=-8\). So the center is \((7,-8)\).

Step7: Solve the sixth question

Given center \((0,-1)\) and radius \(4\). Here \(h = 0,k=-1,r = 4\). Substitute into the formula: \((x-0)^2+(y-(-1))^2=4^2\), which simplifies to \(x^2+(y + 1)^2=16\).

Answer:

1. A. \((x + 2)^2+(y - 1)^2=36\)
2. B. \((3,-2)\)
3. B. \((9,-10)\)
4. C. \(15\)
5. D. \((7,-8)\)
6. D. \(x^2+(y + 1)^2=16\)