if the distances from origin of the centres of 3

Question

If the distances from origin of the centres of 3 circles x²+y²+2alphaix= a² (i=1,2,3) are in G.P. , then length of the tangents drawn to them frm any point on the circles x2+y2 = a2 are in :

(a) A.P (b) G.P (c) H.P (d) None of these


Ans)   C

centres of the 3 circles are (-alphai,0) (i= 1,2,3)

Distances of thes from origin are alpha1, alpha2, alpha 3

now (alpha2)^2= (alpha1)(alpha3)

Also general point on  x²+y²=a² is (a costheta, a sintheta) length of tangent from this point on the given circle is

sqrt( (a cos theta)2+(a sintheta)2 +2alphai(a costheta) -a2)) = sqrt ( 2alphai(a costheta))

now as alpha 1, alpha 2, alpha 3 are in GP

then sqrt alpha 1, sqrt alpha 2, sqrt alpha 3 are in gp

then sqrt 2acos theta alpha 1 , 2acos theta alpha 2 and sqrt 2acos theta alpha 3 are also in GP