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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. is (b) ...plzz explain..
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. is (b) ...plzz explain..
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
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