Question

Which of the following options is correct regarding the straight lines represented by the equation $9{x^2} - 12xy + 4{y^2} = 0$ ?
A. They are coincident.
B. They are perpendicular.
C. They intersect at ${60^ \circ }$ .
D. They are parallel.

Answer 1

Hint: A pair of straight lines, passing through the origin, are represented by a general equation of the form $a{x^2} + 2hxy + b{y^2} = 0$ . 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, passing through the origin, represented by $a{x^2} + 2hxy + b{y^2} = 0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .

Complete step-by-step solution:
Given equation:
$9{x^2} - 12xy + 4{y^2} = 0$ … (1)
General form of a pair of straight lines passing through the origin:
$a{x^2} + 2hxy + b{y^2} = 0$ … (2)
Comparing equation (1) with the general form given in equation (2),
$a = 9$ ,
$b = 4$ and
$h = - 6$
Now, we know that the angle between two lines is calculated using:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {{{\left( { - 6} \right)}^2} - \left( 9 \right)\left( 4 \right)} }}{{9 + 4}}} \right|$
This on further simplification gives:
$\tan \theta = \left| {\dfrac{{2\sqrt {36 - 36} }}{{13}}} \right|$
This gives: $\tan \theta = 0$
Now, this means that the pair of straight lines could be parallel or coincident.
However, since they are both passing through the origin, this means that they cannot be parallel at all as they have a common point between them, which is the origin.
Therefore, they are coincident.
Thus, the correct option is A.

Note: The condition for the lines to be parallel and coincident is the same, that is, ${h^2} - ab$ should be to $0$. However, you should keep in mind that the given equation represents a pair of straight lines passing through the origin. This means that they have at least one common point between them; the origin which means that they cannot be parallel at all. Hence, we can conclude them to be coincident.