Algebraic manipulation is an art

Geometry Level 5

Let a2+b2=1{ a }^{ 2 } + { b }^{ 2 } = 1. such that a,ba,b \in \Re

If f(a,b)=βbαaf\left( a,b \right) = \cfrac { \beta - b }{ \alpha - a } for some fixed pair (α,β)(\alpha ,\beta ) such that α,β>1\alpha, \beta >1

Then maximum and minimum value of f(a,b)f\left(a, b \right) are γ\gamma and δ\delta respectively. And if

P=γ+δ+γ2+δ2P= \gamma + \delta + \sqrt { { \gamma }^{ 2 } + { \delta }^{ 2 } }

Then it can be expressed as:

P=K(αβ)+Lα2β2M(α21)(β21)α21P = \large \frac { K ( \alpha \beta ) + \sqrt { L { \alpha }^{ 2 } { \beta }^{ 2 } - M({ \alpha }^{ 2 }-1)({ \beta }^{ 2 }-1)} }{ { \alpha }^{ 2 }-1 }

Find the value of K+L+MK + L + M

This is part of set Click here

Details and Assumptions

  • Solve it geometrically!

  • K,L,M K , L , M are positive integers.

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