# 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 $$K + L + M$$

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Details and Assumptions

• Solve it geometrically!

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

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