Algebraic manipulation is an art

Geometry Level 5

Let \({ a }^{ 2 } + { b }^{ 2 } = 1\). such that \(a,b \in \Re\)

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

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

\[P= \gamma + \delta + \sqrt { { \gamma }^{ 2 } + { \delta }^{ 2 } } \]

Then it can be expressed as:

\[P = \large \frac { K ( \alpha \beta ) + \sqrt { L { \alpha }^{ 2 } { \beta }^{ 2 } - M({ \alpha }^{ 2 }-1)({ \beta }^{ 2 }-1)} }{ { \alpha }^{ 2 }-1 } \]

Find the value of \(K + L + M \)

This is part of set Click here

Details and Assumptions

  • Solve it geometrically!

  • \( K , L , M \) are positive integers.

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