$\frac{x}{a} \cos \frac{{\alpha \,\, + \,\,\beta }}{2} + \frac{y}{b} \sin \frac{{\alpha \,\, + \,\,\beta }}{2} = cos \frac{{\alpha \,\, - \,\,\beta }}{2}$
Put $ \beta = \alpha + \frac{\pi }{2}$ , equation reduces to,
$bx (cos \alpha - sin \alpha ) + ay (cos \alpha + sin \alpha ) = ab$...$(1)$
compare with $l x + my = - n$.....$(2) $
$\left. {\begin{array}{*{20}{c}} {\cos \,\alpha \,\, - \,\,\sin \,\alpha \,\,\, = \,\,\,{\textstyle{{a\,\ell } \over { - \,n}}}}\\ {\cos \,\alpha \,\, + \,\,\sin \,\alpha \,\,\, = \,\,\,{\textstyle{{m\,b} \over { - \,n}}}} \end{array}} \right\}$Squaring and adding $a^2 l^2 + b^2 m^2 - 2 n^2 = 0 $
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