Q7 of 31 Page 218

Two circles with centers A and B touch each other internally. Another circle touches the larger circle externally at the point x and the smaller circle externally at the point y. If O be the centre of that circle, let us prove that AO + BO is constant.

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Let the radius of the circle with centre A be Ra, B be Rb and O be Ro


Length OA = Radius of circle O + Radius of circle A


⇒ OA = Ro + Ra


Length OB = Radius of circle O + Radius of circle B


⇒ OB = Ro + Rb


⇒AO + BO = Ro + Ra + Ro + Rb


⇒AO + BO = 2Ro + Ra + Rb


Since the radius is always a constant quantity so AO + BO is also a constant quantity.


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