Question

Select the correct option regarding the lines represented by the equation ${x^2} + 2\sqrt 3 xy + 3{y^2} - 3x - 3\sqrt 3 y - 4 = 0$ .
A. They are perpendicular to each other.
B. They are parallel to each other.
C. They are inclined at ${45^ \circ }$ to each other.
D. None of these

Answer 1

Hint: A pair of straight lines are represented by a general equation of the form $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ . Sum of the slopes of the two lines is given by $\dfrac{{ - 2h}}{b}$ and the product of the slopes is given by $\dfrac{a}{b}$ . The angle between the two lines, $\theta $ , is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ . We will use this formula to derive the condition and use it to get the desired solution.

Formula Used: The angle between a pair of straight lines represented by $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .

Complete step-by-step solution:
Given equation of a pair of straight lines:
${x^2} + 2\sqrt 3 xy + 3{y^2} - 3x - 3\sqrt 3 y - 4 = 0$ … (1)
General form of a pair of straight lines passing through the origin:
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ … (2)
Comparing equation (1) with the general form given in equation (2),
$a = 1$ ,
$b = 3$ ,
$c = - 4$ ,
$f = \dfrac{{ - 3\sqrt 3 }}{2}$ ,
$g = - \dfrac{3}{2}$ and
$h = \sqrt 3 $
Now, we know that the tangent of the angle, let’s say $\theta $ , between two lines is calculated using:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values of the variables, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {3 - 3} }}{{1 + 3}}} \right|$
On simplifying further, we get:
$\tan \theta = 0$
This means that the given pair of straight lines could be either parallel or coincident.
From the options provided, option B satisfies the result.
Thus, the correct option is B.

Note: The given options describe the orientation of the two lines, therefore, we have used the formula to evaluate the angle between the two lines. Make sure to avoid any mistakes while comparing the given equation with the general form and while substituting them in the formula to calculate the tangent of the angle. This will prevent any further miscalculations.