# FIND EQUATION OF THE CIRCLES WITH GIVEN INFORMATION

Equation of circle with center (0, 0) :

x2 + y2 = r2

Equation of circle with center (h, k) :

(x - h)2 + (y - k)2 = r2

Distance between center and point on the circle = radius

Midpoint of endpoints of diameter = center

Area of circle = πr2

Circumference of circle = 2πr

Problem 1 :

Ends of diameter : (-17, -9) and (-19, -9).

Solution :

Where (h, k) = -18, -9

(x - h)2 + (y - k)2 = r2

(x + 18)2 + (y + 9)2 = r2 --- (1)

(-17, -9) substitute the equation (1).

(-17 + 18)2 + (-9 + 9)2 = r2

12 = r2

r2 = 1 substitute the equation (1).

(x + 18)2 + (y + 9)2 = 1

Problem 2 :

Ends of diameter : (-3, 11) and (3, -13).

Solution :

Where h = 0 and k = -1

(x - h)2 + (y - k)2 = r2

(x - 0)2 + (y + 1)2 = r2

x2 + (y  + 1)2 = r2--- (1)

(3, -13) substitute the equation (1).

32 + (-13  + 1)2 = r2

9 + (12)2 = r2

153 = r2

r2 = 153 substitute the equation (1).

x2 + (y + 1)2 = 153

Problem 3 :

Center : (-15, 3√7), Area : 2π

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

h, k is  -15, 3√7

(x + 15)2 + (y - 3√7)2 = r2 --- (1)

Area of the circle = πr2

2π = πr2

r2 = 2

r2 = 2 substitute the equation (1).

(x + 15)2 + (y - 3√7)2  = 2

Problem 4 :

Center : (-11, -14), Area : 16π

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

Where h = -11 and k = -14

(x + 11)2 + (y + 14)2 = r2 --- (1)

Area of the circle = πr2

16π = πr2

r2 = 16

r2 = 16 substitute the equation (1).

(x + 11)2 + (y + 14)2  = 16

Problem 5 :

Center : (-5, 12), Circumference : 8π

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

Where h = -5 and k = 12

(x + 5)2 + (y - 12)2 = r2 --- (1)

Area of the circle = 2πr

8π = 2πr

r = 4

r = 4 substitute the equation (1).

(x + 5)2 + (y - 12)2  = 42

(x + 5)2 + (y - 12)2  = 16

Problem 6 :

Center : (15, 14), Circumference : 2π√15

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

Where h = 15 and k = 14

(x - 15)2 + (y - 14)2 = r2 --- (1)

Area of the circle = 2πr

2π√15 = 2πr

r = √15

r = √15 substitute the equation (1).

(x - 15)2 + (y - 14)2  = (√15)2

(x - 15)2 + (y - 14)2  = 15

Problem 7 :

Center : (2, -5), Point on circle : (-7, -1)

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

Where h = 2 and k = -5

(x - 2)2 + (y + 5)2 = r2 --- (1)

Point on circle r = √[(x2 - x1)2 + (y2 - y1)]

(x1, y1) = (2, -5)

(x2, y2) = (-7, -1)

= √[(-7 - 2)2 + (-1 + 5)2]

= √[(-9)2 + (4)2]

=  √[81  +16]

r = √97

r = √96 substitute the equation (1).

(x - 2)2 + (y + 5)2  = (√96)2

(x - 2)2 + (y + 5)2  = 97

Problem 8 :

Center : (14, 17), Point on circle : (15, 17)

Solution :

Equation of circle with center (h ,k) :

(x - h)2 + (y - k)2 = r2

Where h = 14 and k = 17

(x - 14)2 + (y - 17)2 = r2 --- (1)

Point on circle r = √[(x2 - x1)2 + (y2 - y1)2]

(x1, y1) = (14, 17)

(x2, y2) = (15, 17)

= √[(15 - 14)2 + (17 - 17)2]

= √(1)2

r = 1

r = 1 substitute the equation (1).

(x - 14)2 + (y - 17)2  = 12

(x - 14)2 + (y - 17)2  = 1

Problem 9 :

Center : (-11, -8), Radius : 4

Solution :

Center of the circle is (x - h)2 + (y - k)2 = r2

Where h = -11 and k = -8

(x + 11)2 + (y + 8)2 = 42

(x + 11)2 + (y + 8)2 = 16

Problem 10 :

Center : (-6, -15), Radius : √5

Solution :

Center of the circle is (x - h)2 + (y - k)2 = r2

h, k is  -6, -15

(x + 6)2 + (y + 15)2 = (√5)

(x + 6)2 + (y + 15)2 = 5

Problem 11 :

Solution :

By observing the figure,

Center : (0, 3)

Center of the circle is (x - h)2 + (y - k)2 = r2

Where h = 0 and k = 3

(x - 0)2 + (y - 3)2 = 32

x2 + (y - 3)2 = 9

Problem 12 :

Solution :

By observing the figure,

Center : (4, -3)

Center of the circle is (x - h)2 + (y - k)2 = r2

Where h = 4 and k = -3

(x - 4)2 + (y + 3)2 = 12

(x - 4)2 + (y + 3)2 = 1

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