The straight lines represented by (y−mx)2=a21+m2 and (y−nx)2=a21+n2 form a

The straight lines represented by (y−mx)2=a21+m2 are y−mx=±a1+m2 i.e., y−mx=a1+m2-----(1) and y−mx=−a1+m2------(2) Similarly, the straight lines represented by (y−nx)2=a21+n2 are y−nx=a1+n2----(3) and y−nx=−a1+n2----(4)Since the lines (1) and (2) are parallel, the distance between them isa1+m2+a1+m21+m2=|2a|  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.