1.

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

Answer»

SOLUTION :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 inx and y.
So `(2h)^(2)-4ab=0`
`:. h^(2)=ab`
Also , `abc+2ghf-af^(2)-bg^(2)-ch^(2)=0`.


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