Center And Radius

In this page center and radius we are going to see example problems to find the equation of a circle with center and radius is given.

Example 1:

Find the equation of the circle if the center  is (2,-3) and radius is 4 units

Solution:

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

(h,k) = (2,-3) and r = 4

Equation of a circle:

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

                    (x-2)² + (y-(-3))² = 4²

                    (x-2)² + (y+3)² = 16

          x² + 2² - 2(x)(2) + y² + 3² + 2(y)(3) = 16

          x² + 4 - 4x + y² + 9 + 6y = 16

          x² - 4x + y² + 6y + 13 - 16 = 0

          x² - 4x + y² + 6y -3 = 0               

Example 2:

Find the equation of the circle if the center (1,5) and radius 9 and whether the circle passes through the point (2,0)

Solution:

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

(h,k) = (1,5) and r = 9

Equation of a circle:

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

                    (x-1)² + (y-5)² = 9²

          x² + 1² - 2(x)(1) + y² + 5² + 2(y)(5) = 81

          x² + 1 - 2x + y² + 25 + 10y = 81

          x² - 2x + y² + 25y + 26 - 81 = 0

          x² - 2x + y² + 25y -55 = 0               

To check whether the circle passes through the point (2,0) we have to apply the values x=2 and y=0 in the equation of a circle.

          2² - 2(2) + 0² + 25(0) -55 = 0               

            4 - 4 -55 = 0

                    -55 ≠ 0

So the circle is not passing through the point (2,0).

Example 3:

Find the equation of the circle if the center is (1,-3) and passing through the point (4,1)

Solution:

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

But here we have only center of the circle.We don't have the radius.To find tht let us make diagram ith the given details.

To find the radius of a circle we have to find the distance from O to A.

Distance of OA (or) radius of a circle

OA = √(x₂ - x₁)²  + (y₂ - y₁)²

Here  x₁ = 1 x₂ = 4 y₁ = -3  y₂ = 1

OA = √(4 - 1)²  + (1 - (-3))²

OA = √(3)²  + (1 +3)²

OA = √9  + 4²

OA = √9  + 16

OA = √25

OA = √5 x 5

OA = 5 units 

Now we have center and radius

Equation of a circle:

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

                    (x-1)² + (y-(-3))² = 5²

                    (x-1)² + (y+3)² = 25

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

          x² + 1 - 2x + y² + 9 + 6y = 25

          x² - 2x + y² + 6y + 10 - 25 = 0

          x² - 2x + y² + 6y 15 = 0

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