if the distances from the origin of the centres of three circles $$x2+y2+2$$λix−$$c2=0(i=1$$,$$2$$,$$3)$$ are in GP, then the lengths of the tangents drawn to them from any point on the circle $$x2+y2=c2$$ are in

Question

if the distances from the origin of the centres of three circles  $$x2+y2+2$$λix−$$c2=0(i=1$$,$$2$$,$$3)$$  are in GP, then the lengths of the tangents drawn to them from any point on the circle  $$x2+y2=c2$$ are in

Infinity Learn · Accepted Answer

The centres of the given circles are −λi,$$0(i=1$$,$$2$$,$$3)$$  The distances from the origin of the centres are λ$$1$$,λ$$2$$  and λ$$3$$.  It is given that  λ$$22=$$λ$$1$$λ$$3Let$$ $$P(h$$,$$k)$$ be any point on the circle $$x2+y2=c$$ .Then $$h2+k2=c2Now$$,   $$Li=$$  Length of the tangent from (h$$,$$k) to $$x2+y2+2$$λixx−$$c2=0$$⇒ $$Li=h2+k2+2$$λih−$$c2$$⇒$$Li=c2+2$$λih−$$c2$$                                                            ∵$$h2+k2=c2$$⇒$$Li=2$$λjh,    $$i=1$$,$$2$$,$$3$$. ∴   $$L22=2$$λ$$2h$$⇒ $$L22=2h$$λ$$1$$λ$$3$$⇒ $$L22=2$$λ$$1h2$$λ$$3h=L1L3$$                                                                                 ∵λ$$22=$$λ$$1$$λ$$3Hence$$, $$L1$$,$$L2$$,$$L3$$ are in G.P

if the distances from the origin of the centres of three circles x2+y2+2λix−c2=0(i=1,2,3) are in GP, then the lengths of the tangents drawn to them from any point on the circle x2+y2=c2 are in

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