Question

Show that the following equation represents a pair of parallel lines and find distance between them.
(i) $8x^2-24xy+18y^2-6x+9y-5=0$
(ii) $9x^2-6xy+y^2+18x-6y+8=0$
(iii) $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$.

Answer 1

A general equation
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
represent a straight line which are parallel if;-
$h^2=ab$ and $a{f}^2=b{g}^2$
and the distance between them is
$2\sqrt{\frac{g^2-ac}{a(a+b)}}$
$\left(i\right)a=8,h=-12,b=18,g=-3,f=\frac{9}{2},c=-5$
$h^2=ab=144;a{f}^2=162=b{g}^2$
So they represent a pair of parallel at lines
Distance$=2\sqrt{\frac{9+40}{8\times 26}}=\frac{14}{\sqrt{208}}=\frac{7}{2\sqrt{13}}$
$\left(ii\right)a=9,h=-3,b=1,g=9,f=-3,c=8$
$h^2=9=ab;a{f}^2=81=b{g}^2$
Distance$=2\sqrt{\frac{81-72}{90}}=\frac{2}{\sqrt{10}}$
$\left(iii\right)a=1,b=3,h=\sqrt{3},g=-\frac{3}{2},f=-\frac{3\sqrt{3}}{2},c=-4$
$h^2=3=ab;a{f}^2=\frac{27}{4}=b{g}^2$
Distance $=2\sqrt{\frac{\frac{9}{4}+4}{4}}=2\sqrt{\frac{25}{16}}$
$=\frac{5}{2}$