The equation of a circle in standard for is:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

Where the center $\displaystyle (x,y)=(h,k)$ and the radius of the cirlce is $\displaystyle r$.

Dividing by 4 on both sides of the equation yields

$\displaystyle (x-2)^2+(y-3)^2 = \frac{1}{4}$

or

$\displaystyle (x-2)^2+(y-3)^2 = \bigg(\frac{1}{2}\bigg)^2$

an equation whose graph is a circle, centered at (2,3) with radius = .5

Look at the formula for the equation of a circle below. 

$\displaystyle (x - h)^{2}+(y -k)^{2} = r^{2}$

Here $\displaystyle (h, k)$ is the center and $\displaystyle r$ is the radius. Notice that the subtraction in the center is part of the formula. Thus, looking at our equation it is clear that the center is $\displaystyle (3, -4)$ and the radius squared is $\displaystyle 16$. When we square root this value we get that the radius must be $\displaystyle 4$. 

The equation for a circle with center $\displaystyle (h,k)$ and radius $\displaystyle r$ is :

$\displaystyle (x-h)^2+(y-k)^2=r^2$

Our circle is centered at $\displaystyle (3,4)$ with radius $\displaystyle r=1$, so the equation for this circle is :

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

The parent equation of a circle is represented by $\displaystyle \small f(x)=(x-h)^2+(y-k)^2=r^2$. The radius of the circle is equal to $\displaystyle \small r$. The radius of the cirle is $\displaystyle \small \sqrt{36}$.

The equation can be rewritten so that it looks like the parent equation for a circle $\displaystyle \small (x-h)^2+(y-k)^2=r^2$. After completeing the square, the equation changes from $\displaystyle \small x^2+4x+y^2=5$ to $\displaystyle \small x^2+4x+4+y^2=9$. From there it can be expressed as $\displaystyle \small (x+2)^2+y^2=9$. Therefore the center of the circle is at $\displaystyle \small (-2,0)$.

Rewrite the equation of the circle in standard form

$\displaystyle (x - h)^{2} + \left ( y - k\right )^{2} = r^{2}$

as follows:

$\displaystyle x^{2} + y^{2} + 10x - 8y = 84$

$\displaystyle x^{2} + 10x+ y^{2} - 8y = 84$

Since $\displaystyle \left (\frac{10}{2} \right ) ^{2} = 5 ^{2} = 25$ and $\displaystyle \left (\frac{-8}{2} \right ) ^{2} = (-4)^{2} = 16$, we complete the squares by adding:

$\displaystyle x^{2} + 10x+ 25 + y^{2} - 8y + 16 = 84 + 25 + 16$

$\displaystyle (x+5) ^{2} + \left (y- 4 \right ) ^{2} = 125$

The standard form of the equation sets 

$\displaystyle r^{2} = 125$,

so the radius of the circle is 

$\displaystyle r = \sqrt{125} = \sqrt{25} \cdot \sqrt{5} = 5 \sqrt{5}$

The equation of a circle is $\displaystyle (x - h)^2 + (y - k)^2 = r^2$, in which (h, k) is the center of the circle. To derive the center of a circle from its equation, identify the constants immediately following x and y, and flip their signs. In the given equation, x is followed by -1 and y is followed by -6, so the coordinates of the center must be (1, 6).

 To convert the given equation into the format $\displaystyle (x - h)^2 + (y - k)^2 = r^2$, complete the square by adding $\displaystyle (b/2a)^2$ to the x-terms and to the y-terms. 

$\displaystyle x^2 + 6x + y^2 + 2y = -6$

$\displaystyle x^2 + 6x + (6/2)^2 + y^2 + 2y + (2/2)^2 = -6 + (6/2)^2 + (2/2)^2$

$\displaystyle x^2 + 6x + 9 + y^2 + 2y + 1 = -6 + 9 + 1$

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

The square root of 4 is 2, so the radius of the circle is 2. 

 To convert the given equation into the format $\displaystyle (x - h)^2 + (y - k)^2 = r^2$, complete the square by adding $\displaystyle (b/2a)^2$ to the x-terms and to the y-terms. 

$\displaystyle x^2 + 8x + y^2 - 4y = 5$

$\displaystyle x^2 + 8x + (8/2)^2 + y^2 - 4y + (-4/2)^2 = 5 + (8/2)^2 + (-4/2)^2$

$\displaystyle x^2 + 8x + 16 + y^2 - 4y + 4 = 5 + 16 + 4$

$\displaystyle (x + 4)^2) + (y - 2)^2 = 25$

The square root of 25 is 5, so the radius of the circle is 5.

The equation of a circle is $\displaystyle (x - h)^2 + (y - k)^2 = r^2$, in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (0, 0), h and k are both 0. Because the radius is 3, the right side of the equation is equal to 9.