Find The Center Of A Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which is equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius.                                                          

The standard equation of a circle is given by,

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

Where, r is the radius of the circle.

 

Formula for find center and radius

 

If the circle has radius one then it is a unit circle.

                                                Circle with center (h,k) and radius r

The equation of a circle with radius r units and center (h, k) is given by the following equation.

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

This is the standard equation of a circle with (h, k) as it's center and with radius "r" in a XY plane.

If the center of the circle is in the origin, then the previous equation becomes,

h = 0, k = 0

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

(x - 0)2 + (y - 0)2 = r2

x2 + y2 = r2

If the radius of the circle is 1, then the equation becomes

x2 + y2 = 1

This is the equation of a unit circle with origin as center.

Cases:

1) If the center be at the origin and the radius r, then h=0, k=0 and, so the equation of the circle is x2+y2 =r2

2) If the origin lies on the circle, then h2+k2=r2 and, so the equation of the circle in this case is x2+y2- 2hx-2ky=0

3) If the center lies on the x-axis, then the coordinate of the center is zero k=0, and so the equation of the circle is (x-h)2 + y2=r2

4) If the center lies on the x-axis and the origin lies on the circle, then k=0 and h=r, and so the equation of the circle is x2+y2-2rx=0

To find the radius and the center of a circle we use the following formulas,

If the equation of a circle is in the form of

x2 + y2 + 2gx + 2fy + c = 0

Here, the center of the circle is (-g, -f)

and the radius of the circle can be found by the formula,

                 r = `sqrt(g^2 + f^2 -c)` 

 

Examples for finding center and radius

 

Ex 1: Find the center and radius of the circle in the equation, x2 + y2 - 8x - 10y + 4 = 0

Sol:

The given equation is in the form of circle equation x2 + y2 + 2gx + 2fy + c = 0

The center of the circle is (-g, -f)

To find (-g, -f)

2g = -8

g = -4

Also, 2f = -10

f = -5

So the center of the circle is (-4, -5)

To find the radius of the circle, we use the formula   r =  `sqrt(g^2 + f^2 -c)`

We know, g = -4, f = -5, and c = 4

So the radius of the circle, r = `sqrt( (-4)^2 + (-5) ^2 -4)` 

                                                r = `sqrt(16+25 -4) ` 

                                                r = `sqrt(37)`

 So the center and radius of the given circle is (-4, -5) and `sqrt(37)` units

Ex 2: Find the center and radius of the circle x2 + y2 + 4x - 9y + 6 = 0

Sol:

The given equation is in the form of circle equation x2 + y2 + 2gx + 2fy + c = 0

The center of the circle is (-g, -f)

To find (-g, -f)

2g = -4

g = -2

Also, 2f = -9

            f = -9/2

So the center of the circle is (-2, -9/2)

To find the radius of the circle, we use the formula   r =  `sqrt(g^2 + f^2 -c)`

We know, g = -2, f = -9/2, and c = 6

So the radius of the circle, r = `sqrt((-2)^2 + (-9/2)^2 -6)`

                                                r = `sqrt(4+81/4-6)`

                                                r = `sqrt(18.25)` 

                                                r = `sqrt(18.25)` units

So the center and radius of the given circle is (-2, -9/2) and `sqrt(18.25)`  units