Q72 / 125 · BITSAT 2016 · Mathematics
The equation $x^{2}-2 \sqrt{3} x y+3 y^{2}-3 x+3 \sqrt{3} y-4=0 \quad$ represents
- Aa pair of intersecting lines
- Correct answerBa pair of parallel lines with distance between them $\frac{5}{2}$
- Ca pair of parallel lines with distance between them $5 \sqrt{2}$
- Da conic section, which is not a pair of straight lines
Solution
We have $a=1, h=-\sqrt{3}, b=3, g=-\frac{3}{2}$, $\mathrm{f}=\frac{3 \sqrt{3}}{2}, \mathrm{c}=-4$
Thus $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$
Hence the equation represents a pair of straight lines.
Again $\frac{\mathrm{a}}{\mathrm{h}}=\frac{\mathrm{h}}{\mathrm{b}}=\frac{\mathrm{g}}{\mathrm{f}}=-\frac{1}{\sqrt{3}}$
$\therefore$ the lines are parallel. The distance between them
$$
=2 \sqrt{\frac{g^{2}-a c}{a(a+b)}}=2 \sqrt{\frac{\frac{9}{4}+4}{1(1+3)}}=\frac{5}{2}
$$
Hence, the correct answer is option: 'B'.