Given, equation of conic

$a^2{x}^2+b^2{y}^2=a^2\left(a^2+b^2-y^2\right)$

$\Rightarrow a^2{x}^2+b^2{y}^2+a^2{y}^2=a^2\left(a^2+b^2\right)$

$\Rightarrow \frac{a^2{x}^2}{a^2\left(a^2+b^2\right)}+\frac{y^2\left(b^2+a^2\right)}{a^2\left(a^2+b^2\right)}=1$

$\Rightarrow \frac{x^2}{\left(a^2+b^2\right)}+\frac{y^2}{a^2}=1 \dots \left(\mathrm{i}\right)$

$\therefore$Equation $\left(\mathrm{i}\right)$ represents an ellipse whose eccentricity is given by 

$e=\sqrt{1-\frac{a^2}{a^2+b^2}}=\frac{b}{\sqrt{a^2+b^2}}$

Also, we know that in a conic,

Distance of a moving point $P$ from focus $S=e\times$Perpendicular distance of a point $P$ from directrix$\left(PM\right)$

$\Rightarrow SP=ePM$

$\Rightarrow PM=\frac{1}{e}SP=KSP$

$\therefore K=\frac{1}{e}=\frac{\sqrt{a^2+b^2}}{b}$