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.

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 (2, -3), h and k are -2 and 3. Because the radius is 4, the right side of the equation is equal to 16.

The formula for a circle of radius \(\displaystyle r\), centered at the point \(\displaystyle (a,b)\) is given by the general equation:

\(\displaystyle (x-a)^{2} + (y-b^{2}) = r^{2}\)

In this case, the radius is the square root of \(\displaystyle 36\), which is six, and the center is at \(\displaystyle (3, -4).\)

The standard equation for a circle is:
\(\displaystyle \small r^2=(x-h)^2+(y-k)^2\)

\(\displaystyle \small r^2=36\)

\(\displaystyle \small r=6\)

\(\displaystyle \small h=0\)

\(\displaystyle \small k=-2\)

Therefore, the radius is 6 and the center is located at (0,-2)

To find the center or the radius of a circle, first put the equation in the standard form for a circle:  \(\displaystyle (x-x_{1})^2+(y-y_{1})^2=r^2\), where \(\displaystyle r\) is the radius and \(\displaystyle (x_{1},y_{1})\) is the center.

From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula \(\displaystyle (ax-b)^2=a^2x^2-2abx +b^2\) . 

\(\displaystyle a^2x^2=x^2\), so \(\displaystyle a=1\).

\(\displaystyle -2abx=4x\) and \(\displaystyle a=1\), so \(\displaystyle -2b=4\) and \(\displaystyle b=-2\).

Therefore, \(\displaystyle x^2+4x+4=(x+2)^2\).

Because the constant, in this case 4, was not in the original equation, we need to add it to both sides:

\(\displaystyle x^2+4x+y^2-12y=24\)

\(\displaystyle x^2+4x+ 4+y^2-12y=24+4\)

\(\displaystyle (x+2)^2+y^2-12y=28\)

Now we do the same for \(\displaystyle y\):

\(\displaystyle (x+2)^2+y^2-12y+36=28+36\)

\(\displaystyle (x+2)^2+(y-6)^2=64\)

We can now find \(\displaystyle r\):

\(\displaystyle r^2 = 64 = 8\)

To find the center or the radius of a circle, first put the equation in standard form:  \(\displaystyle (x-x_{1})^2+(y-y_{1})^2=r^2\), where \(\displaystyle r\) is the radius and \(\displaystyle (x_{1},y_{1})\) is the center.

From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula \(\displaystyle (ax-b)^2=a^2x^2-2abx +b^2\) . 

\(\displaystyle a^2x^2=x^2\), so \(\displaystyle a=1\).

\(\displaystyle -2abx=-6x\) and \(\displaystyle a=1\), so \(\displaystyle -2b=-6\) and \(\displaystyle b=3\).

This gives \(\displaystyle x^2-6x+9=(x-3)^2\).

Because the constant, in this case 9, was not in the original equation, we must add it to both sides:

\(\displaystyle x^2-6x+y^2+18y=-65\)

\(\displaystyle x^2-6x+9+y^2+18y=-65+9\)

\(\displaystyle (x-3)^2+y^2+18y=-56\)

Now we do the same for \(\displaystyle y\):

\(\displaystyle (x-3)^2+y^2+18y+81=-56+81\)

\(\displaystyle (x-3)^2+(y+9)^2=25\)

We can now find the center: (3, -9)

Remember that the "shifts" involved with circular functions are sort of like those found in parabolas. When you shift a parabola left or right, you have to think "oppositely". A right shift requires you to subtract from the x-component, and a left one requires you to add. Hence, this circle has no horizontal shift, but does shift 6 upward for the vertical component.

You can also remember the general formula for a circle with center at \(\displaystyle {(h,k)}\) and a radius of \(\displaystyle r\).

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

Comparing this to the given equation, we can determine the center point.

\(\displaystyle x^2+(y-6)^2=25\)

\(\displaystyle h=0,\ k=6,\ r=5\)

The center point is at (0,6) and the circle has a radius of 5.

Remember that the shifts for circles work in an opposite manner from what you might think. They are like the parabola's x-component.  Hence, a subtracted variable actually means a shift up or to the right, for the vertical and horizontal components respectively. Since the x-component has a "+5", it is shifted left 5. Since the y-component has a \(\displaystyle -12\), it is shifted upward 12. Therefore, this circle has a center at \(\displaystyle (-5,12)\).

You can also remember the general formula for a circle with center at \(\displaystyle {(h,k)}\) and a radius of \(\displaystyle r\).

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

Comparing this to the given equation, we can determine the center point.

\(\displaystyle (x+5)^2+(y-12)^2=36\)

\(\displaystyle h=-5,\ k=12,\ r=6\)

The center point is at \(\displaystyle (-5,12)\) and the circle has a radius of 6.

Remember that for the equation of a circle, the lone number to the right of the equals sign is the radius squared.

The general formula for a circle with center at \(\displaystyle {(h,k)}\) and a radius of \(\displaystyle r\) is:

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

Comparing this to the given equation, we can determine the radius.

\(\displaystyle (x+5)^2+(y-12)^2=81\)

\(\displaystyle h=-5,\ k=12,\ r=9\)

The center point is at \(\displaystyle (-5,12)\) and the circle has a radius of 9.

Remember that the "shifts" involved with circular functions are sort of like those found in parabolas. When you shift a parabola left or right, you have to think "oppositely". A right shift requires you to subtract from the x-component, and a left one requires you to add. Hence, this circle has a positive 3 horizontal shift, and a negative 2 vertical shift.

You can also remember the general formula for a circle with center at \(\displaystyle {(h,k)}\) and a radius of \(\displaystyle r\).

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

Comparing this to the given equation, we can determine the radius and center point.

\(\displaystyle (x-3)^2+(y+2)^2=49\)

\(\displaystyle h=3,\ k=-2,\ r=7\)

The center point is at \(\displaystyle (3,-2)\) and the circle has a radius of 7.

The question asks us for the sum of these components:

\(\displaystyle 3+(-2)+7=8\)