If the equations of two circles, whose radii are r and R respectively, be S = 0 and S' = 0, then prove that the circles`S/r+-(S')/R=0`will intersect orthogonally

If the equations of two circles whose radii are a and a' are S=0 and S'=0 , then show that the circles S/a+(S')/(a')=0 and S/a-(S')/(a')=0 intersect orthogonally.

The condition for two circles with radii and r_(1) and r_(2) be the distance between their centre,such that the circles are orthogonal is

If S_1=x^2+y^2+2g_1x+2f_1y+c_1=0 and S_2=x^2+y^2+2g_2x +2f_2y+c_2=0 are two circles with radii r_1 and r_2 respectively, show that the points at which the circles subtend equal angles lie on the circle S_1/r_1^2=S_2/r_2^2

If P and Q are conjugate points w.r.t a circle S=x^2+y^2+2gx+2fy+c=0 , then prove that the circle PQ as diameter cuts the circles S=0 orthogonallly.

If P and Q a be a pair of conjugate points with respect to a circle S . Prove that the circle on PQ as diameter cuts the circle S orthogonally.

If P and Q a be a pair of conjugate points with respect to a circle S . Prove that the circle on PQ as diameter cuts the circle S orthogonally.

Let R and S be two relations on a set A. If (ii) R is reflexive and S is any relation prove that R cup S is reflexive