Show that x^2+y^2-6x+4y+2=0 and x^2+y^2+8x+2y-22=0 are orthogonal.

Answer 1

x^2 + y^2 -6x + 4y +2 = 0
can be rewritten as the equation of a circle, as follows.
(x-3)^2 + (y+2)^2 -9 -4 +2 = 0
(x-3)^2 + (y+2)^2 = 11
The center of the circle is (3,-2) and the radius is sqrt(11).
The other equation can be rewritten
(x+4)^2 + (y+1)^2 = 22 -17 = 5
Its center is at (-4,-1) and the radius is sqrt5

It looks to me like the two curves never intersect; I don't see how they can meet the definition of orthogonal.

Answer 2

Solving