Write the General form equations for the following circles:

1.
A circle with center (1, -6) and radius 4
2.
A circle with center (6, 8) and radius 10
3.
A circle with center (0, 3) and radius 2√3
4.
A circle with center (-0.5, 5.5 ) and radius 8.4
5.
A circle with center (a, b) and radius c

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

center at (h,k) radius r

so for example number 3
center at h = 0 and k = 3
radius is 2 sqrt 3 so r^2 = 4*3 = 12
then
(x-0)^2 + (y-3)^2 = 12

or for number 5
(x-a)^2 +(y-b)^2 = c^2

To write the general form equations for circles, we need to use the formula for the equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

Let's use this formula to find the general form equations for the given circles:

1. A circle with center (1, -6) and radius 4:
Using the formula, the equation is (x - 1)^2 + (y + 6)^2 = 4^2.
Expanding this equation, we get x^2 - 2x + 1 + y^2 + 12y + 36 = 16.
Rearranging the terms, the general form equation for this circle is x^2 + y^2 - 2x + 12y + 21 = 0.

2. A circle with center (6, 8) and radius 10:
Using the formula, the equation is (x - 6)^2 + (y - 8)^2 = 10^2.
Expanding this equation, we get x^2 - 12x + 36 + y^2 - 16y + 64 = 100.
Rearranging the terms, the general form equation for this circle is x^2 + y^2 - 12x - 16y - 4 = 0.

3. A circle with center (0, 3) and radius 2√3:
Using the formula, the equation is (x - 0)^2 + (y - 3)^2 = (2√3)^2.
Expanding this equation, we get x^2 + y^2 - 6y + 9 = 12.
Rearranging the terms, the general form equation for this circle is x^2 + y^2 - 6y - 3 = 0.

4. A circle with center (-0.5, 5.5) and radius 8.4:
Using the formula, the equation is (x + 0.5)^2 + (y - 5.5)^2 = 8.4^2.
Expanding this equation, we get x^2 + y^2 + x - 11x + 0.25 + 30.25 = 70.56.
Rearranging the terms, the general form equation for this circle is x^2 + y^2 - 10x - 70.06 = 0.

5. A circle with center (a, b) and radius c:
Using the formula, the equation is (x - a)^2 + (y - b)^2 = c^2.
Expanding this equation, we get x^2 + y^2 - 2ax + 2ay + a^2 + b^2 - c^2 = 0.
Rearranging the terms, the general form equation for this circle is x^2 + y^2 - 2ax + 2ay + (a^2 + b^2 - c^2) = 0.

In the general form equations, the coefficients of x^2 and y^2 are both 1, which is why we don't explicitly write them with a coefficient of 1. The constants in the equation are determined by the values of the center and radius.