The straight lines represented by $(y-mx{)}^2=a^2\left(1+m^2\right)\text{ and }(y-nx{)}^2=a^2\left(1+n^2\right)$ form a

The straight lines represented by

$(y-mx{)}^2=a^2\left(1+m^2\right)\text{ are }y-mx=\pm a\sqrt{1+m^2}$

Since the lines (i) and (ii) are parallel, the distance

between them is $\left|\frac{a\sqrt{1+m^2}+a\sqrt{1+m^2}}{\sqrt{1+m^2}}\right|=\left|2a\right|$

Similarly, the lines (iii) and (iv) are parallel lines and the distance between them is 2a . Since the distances between parallel lines are the same, the four lines form a rhombus