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)

Problem 1

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.

Problem 2

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).

Problem 3

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

1. x² + (y - 13)² = 4

2. (x + 1)² + (y + 12)² = 3

3. (x + 1)² + (y + 3)² = 169

Step-by-step 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

arrange the equations with like terms in columns.

analyze the coefficients of x or y.

add the equations and solve for the remaining variable.

substitute the value into either equation and solve.

check the solution.

step-by-step explanation:

option: d is the correct answer.

d) none of the above.

step-by-step explanation:

clearly from the figure attached to the question we could see that there is no shape given. hence without any shape given in the figure and after rotation we can't obtained a square pyramid.

since, to obtain a image we need a proper figure of the pre-image so that it can be transformed.

hence, the correct answer is:

d) none of the above.

step-by-step explanation: