Q. 73.7( 3 Votes )

# 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.

Answer :

Let the radius of the circle with centre A be R_{a}, B be R_{b} and O be R_{o}

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

⇒ OA = R_{o} + R_{a}

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

⇒ OB = R_{o} + R_{b}

⇒AO + BO = R_{o} + R_{a} + R_{o} + R_{b}

⇒AO + BO = 2R_{o} + R_{a} + R_{b}

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

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