Question

The equation $x^2-2\sqrt{3}xy+3y^2-3x+3\sqrt{3}y-4=0$ represents

• A. a pair of intersecting lines
• B. a pair of parallel lines with distance between them $\frac{5}{2}$
• C. a pair of parallel lines with distance between them $5\sqrt{2}$
• D. a conic section, which is not a pair of straight lines

Answer 1

Answer: (B)

We have $a=1,h=-\sqrt{3},b=3$,
$g=-\frac{3}{2},f=\frac{3\sqrt{3}}{2},c=-4$ .
Thus $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$
Hence the equation represents a pair of straight lines.
Again $\frac{a}{h}=\frac{h}{b}=\frac{g}{f}=-\frac{1}{\sqrt{3}}$
$\therefore$ the lines are parallel. The distance between them
$=2\sqrt{\frac{g^2-ac}{a\left(a+b\right)}}=2\sqrt{\frac{\frac{9}{4}+4}{1\left(1+3\right)}}=\frac{5}{2}$