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 $K + L + M$

**Details and Assumptions**

Solve it geometrically!

$K , L , M$ are positive integers.