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Inside a semi-circle of radius 1 unit, two circles of radii r₁ and r₂ are drawn, each touching the circumference and the diameter of the semi-circle also touches each other externally. Then [max.(r₁ + r₂)] where [x] denotes integral value nearest to x .............

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Detailed Solution

Centres of the smaller circles be $C_{1} (a, r_{1}), C_{2} (b, r_{2})$

${OC}_{1} = R - r_{1} \Rightarrow 1 - a^{2} = 2 r_{1}$

Also, $1 - b^{2} = 2 r_{2}$

and $C_{1} C_{2} = r_{1} + r_{2}$

$\Rightarrow (a - b)^{2} = 4 r_{1} r_{2}$

Also, $r_{1} + r_{2} = 1 - \frac{a^{2} + b^{2}}{2} < 1 - | a | |b|$

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