Equation of a Circle

Standard form equation of a Circle

What is the standard form equaton of a circle?

Answer : is a way to express the definition of a circle on the coordinate plane.

The formula is $$(x -h)^2 + (y - k)^2 =r^2 $$.

  • h and k are the x and y coordinates of the center of the circle
  • $$(x-9)^2 + (y-6)^2 =100 $$ is a circle centered at (9, 6) with a radius of 10

Diagrams

Diagram 1
General Formula general formula equation of circle
Diagram 2
Circle with a center of (4,3) and a radius of 5 Example 1

Another Example

Circle with a center of (2, -1) and a radius of 4 example 2

Practice Problems

Practice 1

What is the equation of the circle pictured on the graph below?

Picture of Equation of circle in standard form

Since the radius of this this circle is 1, and its center is the origin, this picture's equation is.

$$ (y-0)^2 + (x-0)^2 = 1^2 \\ y^2 + x^2 = 1 $$

Practice 2

Look at the graph below, can you express the equation of the circle in standard form?

Picture of Equation of circle in standard form

Since the radius of this this circle is 1, and its center is (1, 0), this circle's equation is.

$$ (y-0)^2 +(x-1)^2 = 1^2 \\ y^2 + (x-1)^2 = 1 $$

Practice 3

Look at the graph below, can you express the equation of the circle in standard form?

Picture of Equation of circle in standard form

Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is.

$$ (x-3)^2 +(y-1)^2 = 2^2 \\ (x-3)^2 +(y-1)^2 = 4 $$

Practice 4

What is the radius of the circles below?

Y2 + X2 = 9

$$ \sqrt{9} =3$$

Y2 + X2 = 16

$$ \sqrt{16} = 4$$

Y2 + X2 = 25

$$ \sqrt{ 25 } = 5 $$

Y2 + X2 = 11

$$ \sqrt{11}$$

Y2 + X2 = a

$$ \sqrt{a}$$

Practice 5

Look at each standard form equation below and identify the center and radius.

r = radius

(y - 3)2 + (x - 1)2 = 9

(1, 3) r = 3

(y - 5)2 + (x - 14)2 = 16

(14, 5) r = 4

(y - 1)2 + (x - 5)2 = 25

(5, 1) r = 5

(x + 2)2 + (y - 12)2 = 36

(-2, 12) r = 6

(y + 7)2 + (x + 5)2 = 49

(-5, -7) r = 7

(x + 8)2 + (y + 17)2 = 49

(-8, -17) r = 7

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