We will proceed to convert the equations into standard format to determine the solution.
we know that
The Standard Form Equation of a Circle is equal to
![(x-h)^{2} +(y-k)^{2} =r^{2}](https://tex.z-dn.net/?f=%28x-h%29%5E%7B2%7D%20%2B%28y-k%29%5E%7B2%7D%20%3Dr%5E%7B2%7D)
where
(h,k) is the center of the circle
r is the radius of the circle
<u>Case N
</u>
![x^{2}+y^{2}-2x+ 2y- 1= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2By%5E%7B2%7D-2x%2B%202y-%201%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2}-2x)+ (y^{2}+ 2y)=1](https://tex.z-dn.net/?f=%28x%5E%7B2%7D-2x%29%2B%20%28y%5E%7B2%7D%2B%202y%29%3D1)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2}- 2x+1)+ (y^{2}+ 2y+1)=1+1+1](https://tex.z-dn.net/?f=%28x%5E%7B2%7D-%202x%2B1%29%2B%20%28y%5E%7B2%7D%2B%202y%2B1%29%3D1%2B1%2B1)
Rewrite as perfect squares
![(x-1)^{2}+(y+1)^{2}=3](https://tex.z-dn.net/?f=%28x-1%29%5E%7B2%7D%2B%28y%2B1%29%5E%7B2%7D%3D3)
<u>Case N
</u>
![x^{2}+ y^{2}-4x + 4y- 10= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D-4x%20%2B%204y-%2010%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2} - 4x)+ (y^{2}+ 4y)=10](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20-%204x%29%2B%20%28y%5E%7B2%7D%2B%204y%29%3D10)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2} - 4x+4)+ (y^{2}+ 4y+4)=10+4+4](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20-%204x%2B4%29%2B%20%28y%5E%7B2%7D%2B%204y%2B4%29%3D10%2B4%2B4)
Rewrite as perfect squares
![(x-2)^{2}+ (y+2)^{2}=18](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%2B%20%28y%2B2%29%5E%7B2%7D%3D18)
![(x-2)^{2}+ (y+2)^{2}=\sqrt{18}^{2}](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%2B%20%28y%2B2%29%5E%7B2%7D%3D%5Csqrt%7B18%7D%5E%7B2%7D)
<u>Case N
</u>
![x^{2}+ y^{2}-8x - 6y-20= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D-8x%20-%206y-20%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2}- 8x)+ (y^{2} - 6y)=20](https://tex.z-dn.net/?f=%28x%5E%7B2%7D-%208x%29%2B%20%28y%5E%7B2%7D%20-%206y%29%3D20)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2}- 8x+16)+ (y^{2}- 6y+9)=20+16+9](https://tex.z-dn.net/?f=%28x%5E%7B2%7D-%208x%2B16%29%2B%20%28y%5E%7B2%7D-%206y%2B9%29%3D20%2B16%2B9)
Rewrite as perfect squares
![(x-4)^{2}+ (y-3)^{2}=45](https://tex.z-dn.net/?f=%28x-4%29%5E%7B2%7D%2B%20%28y-3%29%5E%7B2%7D%3D45)
![(x-4)^{2}+ (y-3)^{2}=\sqrt{45}^{2}](https://tex.z-dn.net/?f=%28x-4%29%5E%7B2%7D%2B%20%28y-3%29%5E%7B2%7D%3D%5Csqrt%7B45%7D%5E%7B2%7D)
<u>Case N
</u>
![4x^{2}+4y^{2}+16x +24y- 40= 0](https://tex.z-dn.net/?f=4x%5E%7B2%7D%2B4y%5E%7B2%7D%2B16x%20%2B24y-%2040%3D%200)
Simplify divide by
both sides
![x^{2}+ y^{2}+4x+6y- 10= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D%2B4x%2B6y-%2010%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2} +4x)+ (y^{2} + 6y)=10](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20%2B4x%29%2B%20%28y%5E%7B2%7D%20%2B%206y%29%3D10)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2}+4x+4)+(y^{2} + 6y+9)=10+4+9](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%2B4x%2B4%29%2B%28y%5E%7B2%7D%20%2B%206y%2B9%29%3D10%2B4%2B9)
Rewrite as perfect squares
![(x+2)^{2}+ (y+3)^{2}=23](https://tex.z-dn.net/?f=%28x%2B2%29%5E%7B2%7D%2B%20%28y%2B3%29%5E%7B2%7D%3D23)
![(x+2)^{2}+ (y+3)^{2}=\sqrt{23}^{2}](https://tex.z-dn.net/?f=%28x%2B2%29%5E%7B2%7D%2B%20%28y%2B3%29%5E%7B2%7D%3D%5Csqrt%7B23%7D%5E%7B2%7D)
<u>Case N
</u>
![5x^{2}+ 5y^{2}-20x +30y+ 40= 0](https://tex.z-dn.net/?f=5x%5E%7B2%7D%2B%205y%5E%7B2%7D-20x%20%2B30y%2B%2040%3D%200)
Simplify divide by
both sides
![x^{2}+ y^{2}-4x +6y + 8= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D-4x%20%2B6y%20%2B%208%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2} -4x)+ (y^{2}+ 6y)=-8](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20-4x%29%2B%20%28y%5E%7B2%7D%2B%206y%29%3D-8)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2}-4x+4)+ (y^{2}+ 6y+9)=-8+4+9](https://tex.z-dn.net/?f=%28x%5E%7B2%7D-4x%2B4%29%2B%20%28y%5E%7B2%7D%2B%206y%2B9%29%3D-8%2B4%2B9)
Rewrite as perfect squares
![(x-2)^{2} + (y+3)^{2}=5](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%20%2B%20%28y%2B3%29%5E%7B2%7D%3D5)
![(x-2)^{2} + (y+3)^{2}=\sqrt{5}^{2}](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%20%2B%20%28y%2B3%29%5E%7B2%7D%3D%5Csqrt%7B5%7D%5E%7B2%7D)
<u>Case N
</u>
![2x^{2} + 2y^{2}-28x -32y-8= 0](https://tex.z-dn.net/?f=2x%5E%7B2%7D%20%2B%202y%5E%7B2%7D-28x%20-32y-8%3D%200)
Simplify divide by
both sides
![x^{2} + y^{2} -14x-16y-4= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B%20y%5E%7B2%7D%20-14x-16y-4%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2} -14x)+ (y^{2} -16y)=4](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20-14x%29%2B%20%28y%5E%7B2%7D%20-16y%29%3D4)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2} -14x+49)+ (y^{2} -16y+64)=4+49+64](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20-14x%2B49%29%2B%20%28y%5E%7B2%7D%20-16y%2B64%29%3D4%2B49%2B64)
Rewrite as perfect squares
![(x-7)^{2} + (y-8)^{2}=117](https://tex.z-dn.net/?f=%28x-7%29%5E%7B2%7D%20%2B%20%28y-8%29%5E%7B2%7D%3D117)
![(x-7)^{2} + (y-8)^{2}=\sqrt{117}^{2}](https://tex.z-dn.net/?f=%28x-7%29%5E%7B2%7D%20%2B%20%28y-8%29%5E%7B2%7D%3D%5Csqrt%7B117%7D%5E%7B2%7D)
<u>Case N
</u>
![x^{2}+ y^{2}+12x - 2y- 9 = 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D%2B12x%20-%202y-%209%20%3D%200)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![(x^{2} +12x)+ (y^{2} - 2y)=9](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20%2B12x%29%2B%20%28y%5E%7B2%7D%20-%202y%29%3D9)
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
![(x^{2} +12x+36)+ (y^{2} - 2y+1)=9+36+1](https://tex.z-dn.net/?f=%28x%5E%7B2%7D%20%2B12x%2B36%29%2B%20%28y%5E%7B2%7D%20-%202y%2B1%29%3D9%2B36%2B1)
Rewrite as perfect squares
![(x+6)^{2} + (y-1)^{2}=\sqrt{46}^{2}](https://tex.z-dn.net/?f=%28x%2B6%29%5E%7B2%7D%20%2B%20%28y-1%29%5E%7B2%7D%3D%5Csqrt%7B46%7D%5E%7B2%7D)
the circles in ascending order of their radius lengths is
N ![1](https://tex.z-dn.net/?f=1)
![x^{2}+ y^{2}-2x + 2y- 1=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D-2x%20%2B%202y-%201%3D0)
N ![2](https://tex.z-dn.net/?f=2)
![5x^{2}+ 5y^{2}-20x +30y+40=0](https://tex.z-dn.net/?f=5x%5E%7B2%7D%2B%205y%5E%7B2%7D-20x%20%2B30y%2B40%3D0)
![(x-2)^{2}+(y+3)^{2}=\sqrt{5}^{2}](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%2B%28y%2B3%29%5E%7B2%7D%3D%5Csqrt%7B5%7D%5E%7B2%7D)
N ![3](https://tex.z-dn.net/?f=3)
![x^{2}+y^{2}-4x+4y- 10=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2By%5E%7B2%7D-4x%2B4y-%2010%3D0)
![(x-2)^{2}+(y+2)^{2}=\sqrt{18}^{2}](https://tex.z-dn.net/?f=%28x-2%29%5E%7B2%7D%2B%28y%2B2%29%5E%7B2%7D%3D%5Csqrt%7B18%7D%5E%7B2%7D)
N ![4](https://tex.z-dn.net/?f=4)
![4x^{2}+4y^{2}+16x+24y-40=0](https://tex.z-dn.net/?f=4x%5E%7B2%7D%2B4y%5E%7B2%7D%2B16x%2B24y-40%3D0)
![(x+2)^{2}+(y+3)^{2}=\sqrt{23}^{2}](https://tex.z-dn.net/?f=%28x%2B2%29%5E%7B2%7D%2B%28y%2B3%29%5E%7B2%7D%3D%5Csqrt%7B23%7D%5E%7B2%7D)
N ![5](https://tex.z-dn.net/?f=5)
![x^{2}+y^{2}-8x- 6y-20= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2By%5E%7B2%7D-8x-%206y-20%3D%200)
![(x-4)^{2}+(y-3)^{2}=\sqrt{45}^{2}](https://tex.z-dn.net/?f=%28x-4%29%5E%7B2%7D%2B%28y-3%29%5E%7B2%7D%3D%5Csqrt%7B45%7D%5E%7B2%7D)
N ![6](https://tex.z-dn.net/?f=6)
![x^{2}+ y^{2}+12x- 2y-9= 0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B%20y%5E%7B2%7D%2B12x-%202y-9%3D%200)
![(x+6)^{2}+(y-1)^{2}=\sqrt{46}^{2}](https://tex.z-dn.net/?f=%28x%2B6%29%5E%7B2%7D%2B%28y-1%29%5E%7B2%7D%3D%5Csqrt%7B46%7D%5E%7B2%7D)
N ![7](https://tex.z-dn.net/?f=7)
![2x^{2}+2y^{2}-28x-32y-8=0](https://tex.z-dn.net/?f=2x%5E%7B2%7D%2B2y%5E%7B2%7D-28x-32y-8%3D0)
![(x-7)^{2}+(y-8)^{2}=\sqrt{117}^{2}](https://tex.z-dn.net/?f=%28x-7%29%5E%7B2%7D%2B%28y-8%29%5E%7B2%7D%3D%5Csqrt%7B117%7D%5E%7B2%7D)