If the equation `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` respresent the pair of parallel straight lines , then prove that `h^(2)=abandabc+2fgh-af^(2)-bg^(2)-ch^(2)=0`.

Explanation

Let the given equation represent pair of parallel straigh lines `lx+my+n=0andlx+my+n'=0`
`:. ax^(2)+2hxy+b^(2)+2gx+2fy+c`
`=(lx+my+n)(lx+my+n')`
`=(lx+my)^(2)+l(n+n')x+m(n+n')y+nn'=0`
Thus , expression `(lx+my)^(2)` is same as `ax^(2)+by^(2)_2hxy`.
Therefore , `ax^(2)+2hxy+by^(2)` must be perfect square of liner expression in x and y.
So `(2h)^(2)-4ab=0`
`:. h^(2)=ab`
Also , `abc+2ghf-af^(2)-bg^(2)-ch^(2)=0`.