Question:

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

Updated On: Aug 15, 2022
  • a pair of intersecting lines
  • a pair of parallel lines with distance between them $\frac{5}{2}$
  • a pair of parallel lines with distance between them $5 \sqrt{2}$
  • a conic section, which is not a pair of straight lines
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The Correct Option is B

Solution and Explanation

We have $a = 1, h = - \sqrt{3} , b = 3$, $ g = -\frac{3}{2} , f = \frac{3 \sqrt{3}}{2} , c = - 4$ . Thus $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ 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} $
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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c