Question

Part 2: Use the information provided to write the standard equation of each circle. Please show work.

1. x^2 + y^2 - 26y + 165 = 0

2. x^2 + y^2 + 2x + 24y + 142 = 0

3. Ends of a diameter: (-1, -16), (-1, 10)

Answer 1

1. x² + (y - 13)² = 4

2. (x + 1)² + (y + 12)² = 3

3. (x + 1)² + (y + 3)² = 169

Explanation:

1. x^2 + y^2 - 26y + 165= 0

h = 0/-2 = 0

k = -26/-2 = 13

0² + 13² - 165 = r²

r² = 4

r = 2

(x - 0)² + (y - 13)² = 2²

x² + (y - 13)² = 4

2. x^2 + y^2 + 2x + 24y + 142= 0

h = 2/-2 = -1

k = 24/-2 = -12

(-1)² + (-12)² - 142 = r²

r² = 3

(x - (-1))² + (y - (-12))² = 3

(x + 1)² + (y + 12)² = 3

15. Ends of a diameter: (-1, -16) and (-1, 10)

Centre: midpoint of the diameter

(h,k) = (-1-1)/2, (-16+10)/2

= (-1,-3)

Diameter length= 16 - -10 = 26

Radius = 26/2 = 13

(x - (-1))² + (y - (-13))²= 13²

(x + 1)² + (y + 3)² = 169

Answer 2

Problem 1

Answer: x^2 + (y - 13)^2 = 4

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Work Shown:

x^2 + y^2 - 26y + 165 = 0

x^2 + (y^2 - 26y) + 165 = 0

x^2 + (y^2 - 26y + 169) - 169 + 165 = 0 ... see note below

x^2 + (y - 13)^2 - 4 = 0

x^2 + (y - 13)^2 = 4

This is a circle with center (0,13) and radius 2

note: I took half of the y coefficient 26 to get 13, then I squared 13 to get 169. This is added and subtracted to help keep the equation balanced.

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Problem 2

Answer: (x+1)^2 + (y+12)^2 = 3

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Work Shown:

x^2 + y^2 + 2x + 24y + 142 = 0

(x^2 + 2x) + (y^2 + 24y) + 142 = 0

(x^2 + 2x + 1) - 1 + (y^2 + 24y + 144) - 144 + 142 = 0

(x+1)^2 + (y+12)^2 - 3 = 0

(x+1)^2 + (y+12)^2 = 3

This is a circle with center (-1,-12). The radius is sqrt(3).

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Problem 3

Answer: (x+1)^2 + (y+3)^2 = 169

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Work Shown:

Use the midpoint formula to find the center of the circle

xm = (x1+x2)/2

xm = (-1+(-1))/2

xm = -1

ym = (y1+y2)/2

ym = (-16+10)/2

ym = -3

The midpoint is (xm,ym) = (-1,-3)

The center of the circle is (h,k) = (-1,-3)

Use the distance formula to find the distance between the two given endpoints

d = sqrt((x1 - x2)^2 + (y1 - y2)^2)

d = sqrt((-1-(-1))^2 + (-16-10)^2)

d = sqrt((-1+1)^2 + (-16-10)^2)

d = sqrt((0)^2 + (-26)^2)

d = sqrt(0 + 676)

d = sqrt(676)

d = 26

The distance between the two endpoints is 26, so the diameter is 26

The radius is half that at r = d/2 = 26/2 = 13

Now we plug the center and radius into the standard equation of a circle

(x-h)^2 + (y-k)^2 = r^2

(x-(-1))^2 + (y-(-3))^2 = 13^2

(x+1)^2 + (y+3)^2 = 169