# FIND EQUATION OF CIRCLE FROM GRAPH

To find equation of circle, we need two quantities.

i) Center of the circle

When we have these two information, using the formula given below, we can find equation of the circle.

x2 + y2 = r2

Center is (0, 0) and radius is r.

(x - h)2 + (y - k)2 = r2

Center is (h, k) and radius is r.

Write down the equations of each of the circles shown below :

Problem 1 :

Solution :

Centre = (0, 0)

Equation of a circle is x2 + y2 = r2

x2 + y2 = 22

x2 + y2 = 4

So, equation of a circle is x2 + y2 = r2.

Problem 2 :

Solution :

Centre = (1, 0) and Radius r = 1

(h, k) = (1, 0)

Equation of a circle is (x – h)2 + (y – k)2 = r2

(x – 1)2 + (y – 0)2 = 1

x2 + 12 - 2(x)(1) + y2 = 1

x2 + 1 - 2x + y2 = 1

x2 + y2 - 2x + 1 = 1

x2 + y2 - 2x = 0

So, equation of a circle is x2 + y2 – 2x = 0.

Problem 3 :

Solution :

Centre = (3, 3) and Radius r = 3

(h, k) = (3, 3)

Equation of a circle is (x – h)2 + (y – k)2 = r2

(x – 3)2 + (y – 3)2 = 32

x2 + 32 - 2(x)(3) + y2 + 32 - 2(y)(3) = 9

x2 + 9 – 6x + y2 + 9 - 6y = 9

x2 + y2 – 6x – 6y + 18 = 9

Subtract 9 from both sides.

x2 + y2 – 6x – 6y + 18 - 9 = 9 – 9

x2 + y2 – 6x – 6y + 9 = 0

So, equation of a circle is x2 + y2 – 6x – 6y + 9 = 0.

Problem 4 :

Solution :

Centre = (-1, -1) and Radius r = 1

(h, k) = (-1, -1)

Equation of a circle is (x – h)2 + (y – k)2 = r2

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

x2 + 12 + 2(x)(1) + y2 + 12 + 2(y)(1) = 1

x2 + 1 + 2x + y2 + 1 + 2y = 1

x2 + y2 + 2x + 2y + 2 = 1

x2 + y2 + 2x + 2y + 1 = 0

So, equation of a circle is x2 + y2 + 2x + 2y + 1 = 0.

Write down the equations of the following circles :

Problem 5 :

Solution :

Centre = (3, -1) and Radius r = 5

(h, k) = (3, -1)

Equation of a circle is (x – h)2 + (y – k)2 = r2

(x – 3)2 + (y + 1)2 = 52

x2  + 32 – 2(x)(3) + y2 + 12 + 2(y)(1) = 25

x2 + 9 – 6x + y2 + 1 + 2y = 25

x2 + y2 - 6x + 2y + 10 = 25

x2 + y2 - 6x + 2y - 15 = 0

So, equation of a circle is x2 + y2 - 6x + 2y - 15 = 0.

Problem 6 :

Solution :

Centre = (0, 6) and Radius r = 6

(h, k) = (0, 6)

Equation of a circle is (x – h)2 + (y – k)2 = r2

(x – 0)2 + (y - 6)2 = 62

x2  + 02 – 2(x)(0) + y2 + 62 - 2(y)(6) = 36

x2 + 0 – 0x + y2 + 36 - 12y = 36

x2 + y2 - 12y + 36 = 36

x2 + y2 - 12y = 0

So, equation of a circle is x2 + y2 - 12y = 0.

Problem 7 :

Solution :

Centre = (4, 5) and Radius r = 4

(h, k) = (4, 5)

Equation of a circle is (x – h)2 + (y – k)2 = r2

(x – 4)2 + (y - 5)2 = 42

x2  + 42 – 2(x)(4) + y2 + 52 - 2(y)(5) = 16

x2  + 16 – 8x + y2 + 25 - 10y = 16

x2  – 8x + y2 - 10y + 16 - 16 + 25 = 0

x2  – 8x + y2 - 10y + 25 = 0

So, equation of a circle is x2 + y2 – 8x - 10y + 25 = 0.

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