Question

Part 3: Use the information provided to write the standard equation of each circle. Please show the work.

1. Ends of a diameter: (-5, 5) and (-5, 7)

2. Ends of a diameter: (2, -9) and (10, -9)

3. Ends of a diameter: (-9, 14) and (-9, -8)

Answer 1

16. (x + 5)² + (y - 6)² = 1

17. (x - 6)² + (y + 9)² = 16

18. (x + 9)² + (y - 3)² = 121

Explanation:

Circle:

(x - h)² + (y - k)² = r²

16. Ends of a diameter: (-5, 5) and (-5, 7)

(h,k) = (-5-5)/2, (5+7)/2 = (-5,6)

Diameter = 7 - 5 = 2

Radius = 2/2 = 1

(x - (-5))² + (y - 6)² = 1²

(x + 5)² + (y - 6)² = 1

17. Ends of a diameter: (2, -9) and (10, -9)

(h,k) = (2+10)/2,(-9-9)/2 = (6,-9)

Diameter = 10 - 2 = 8

Radius = 8/2 = 4

(x - 6)² + (y - (-9))² = 4²

(x - 6)² + (y + 9)² = 16

18. Ends of a diameter: (-9, 14) and (-9, -8)

(h,k) = (-9-9)/2, (14-8)/2 = (-9,3)

Diameter = 14 - (-8) = 22

Radius = 22/2 = 11

(x - (-9))² + (y - 3)² = 11²

(x + 9)² + (y - 3)² = 121

Answer 2

Answer: 1. (x + 5)² + (y - 6)² = 1

2. (x - 6)² + (y + 9)² = 16

3. (x + 9)² + (y - 3)² = 121

Explanation:

The standard equation of a circle is (x - h)² + (y - k)² = r²

• (h, k) is the Center
• The center is the midpoint of the endpoints
• r is the radius
• radius is the distance from the center to the endpoint

1) endpoints: (-5, 5) and (-5, 7) → h = -5

`k=(5+7)/(2)=(12)/(2)=6\\\\\\r=|Center-endpoint|\\.\ =|6-5|\\.\ =1`

Now input (h, k) = (-5, 6) and r = 1 into the standard equation

(x + 5)² + (y - 6)² = 1² → (x + 5)² + (y - 6)² = 1

*********************************************************************************************

2) endpoints: (2, -9) and (10, -9) → k = -9

`h=(2+10)/(2)=(12)/(2)=6\\\\\\r=|Center-endpoint|\\.\ =|6-2|\\.\ =4`

Now input (h, k) = (6, -9) and r = 4 into the standard equation

(x - 6)² + (y + 9)² = 4² → (x - 6)² + (y + 9)² = 16

*********************************************************************************************

3) endpoints: (-9, 14) and (-9, -8) → h = -9

`k=(14+(-8))/(2)=(6)/(2)=3\\\\\\r=|Center-endpoint|\\.\ =|3-(-8)|\\.\ =11`

Now input (h, k) = (-9, 3) and r = 11 into the standard equation

(x + 9)² + (y - 3)² = 11² → (x + 9)² + (y - 3)² = 121