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

HTML Comment Box is loading comments...

                                        Home





Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.




Featured Categories 

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test