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

The correct option is B A pair of parallel lines with distance between them $\frac{5}{2}$

Given equation is $x^2-2\sqrt{3}xy+3y^2-3x+23\sqrt{3}y-4=0$

We have $a=1,h=-\sqrt{3},g=-\frac{3}{2},f=\frac{3\sqrt{3}}{2},c=-4$

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

$=2\sqrt{\frac{g^2-ac}{a(a+b)}}=2\sqrt{\frac{\frac{9}{4}+4}{1(1+3)}}=\frac{5}{2}$

Hence, option B is correct.