Question

How do i find the center and radius of the given circle

Answer 1

There are multiple ways to find the center and radius of a circle, depending on what information your provided.

Method 1: Reading the standard equation of a circle

Equation of a circle:
`(x-h)^(2) +(y-k)^(2) =r^(2)`

where h is the center's x-coordinate, k is the center's y-coordinate, and r is the radius. Here's an example:

`(x-5)^(2) +(y+2)^(2) =25`

In this case, the center would be (5,-2) and the radius is 5

Method 2: Reading a graph

In the attached graph, there is a circle with a labeled center point. The radius can be found by finding the distance from the center to a different point on the circle.

Distance Formula:

`D = \sqrt{ (x_(2)-x_(1)) ^(2) + (y_(2)-y_(1)) ^(2)}\\`

The first point we use will be the center ( 5 , -2 ), and the second point will be a point on the circle, (10,-2).

`r = D = \sqrt{ ((10)-(5)) ^(2) + ((-2)-(-2) )^(2)}\\r = D = \sqrt{ (5 ^(2) + 0)^(2)}\\r = D = √(25)\\r = 5`

Next, plug in the values into the standard equation of a circle formula

`(x-5)^(2) +(y+2)^(2) =5^2`

Simplify

`(x-5)^(2) +(y+2)^(2) =25`