If circle S1 has radius R1 and circle S2 has radius R2 then prove that circles

[S1/R1 + S2/R2]and [S1/R1 - S2/R2] are orthogonal

Dear Student,

        Let us take a general case of a circle.

        Say the equation of the circle s, is

                x² + y² + 2gx + 2fy + c = 0

                where radius R₁ =  (g2 + f2 - c)^1/2

        and say S₂ is

                x² + y² + 2g¹x + 2f¹y + c¹ = 0

        where radius R₂ = (g¹² + f¹² - c¹)^1/2

        Now in order to prove that the curves (S₁/R₁ + S₂/R₂ )

and (S₁/R₁ + S₂/R₂ ) cut each other orthogonally we have to show that dy/dx  to the first curve at the point of intersection and dy/dx to the second curve at the same point when multiplied = –1

        i.e., (dy/dx)_c1 * (dy/dx)_c2 = –1

Note : Students are advised to remember this property as a result.