Question

Show that the circles `4x^2+4y^2-10x+16y+2=0`,and `4x^2+4y^2+12x+6y-26=0` are orthogonal?

Answer 1

The reference Orthogonal Circles says that circles:
`x^2+y^2+2gx+2fy+c=0`
`x^2+y^2+2g'x+2f'y+c'=0`
Are orthogonal if
`2gg'+2ff'=c+c'`

Explanation:

Given:

`4x^2+4y^2-10x+16y+2=0" [1]"`
`4x^2+4y^2+12x+6y-26=0" [2]"`

To force equations [1] and [2] to be in the same form as the equations given by the reference, divide them by 4:

`x^2+y^2-5/2x+4y+1/2=0" [1.1]"`
`x^2+y^2+3x+3/2y-13/2=0" [2.1]"`

Please observe that in equation [1.1]:

#2g = -5/2, 2f = 4, and c = 1/2

Solving for g and f:

`g = -5/4, f = 2, and c = 1/2`

A similar observation from equation [2.1]:

`2g' = 3, 2f' = 3/2, and c' = -13/2`

Solving for g' and f':

`g' = 3/2, f' = 3/4, and c' = -13/2`

Substitute these values into the equation:

`2gg'+2ff'=c+c'`

`2(-5/4)(3/2)+2(2)(3/4)=1/2-13/2`

`-15/4+3= -12/2`

`-3/4 != -6`

They are not orthogonal.