Question

If the lines $\left( {p - q} \right){x^2} + 2\left( {p + q} \right)xy + \left( {q - p} \right){y^2} = 0$ are perpendicular, then which of the following options is correct?
A. $p = q$
B. $p = 0$
C. $q = 0$
D. $p$ and $q$ may have any value

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:
Let us consider a general pair of straight lines, passing through the origin.
$a{x^2} + 2hxy + b{y^2} = 0$ … (1)
Let the angle between them be $\theta $ .
Now, we know that the tangent of the angle between them is given by the formula:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ … (2)
For the lines to be perpendicular to each other, $\theta = \dfrac{\pi }{2}$ .
Substituting this in equation (2), we get:
$\tan \dfrac{\pi }{2} = \infty = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
From this we can conclude that:
$a + b = 0$
This gives:
$a = - b$ … (3)
Hence, for a pair of straight lines to be perpendicular to each other, $a = - b$ .
Given equation:
$\left( {p - q} \right){x^2} + 2\left( {p + q} \right)xy + \left( {q - p} \right){y^2} = 0$
Comparing the above equation with the general form, given in (1), we get:
$a = p - q,b = q - p$
Therefore, $a = - b$ .
From equation (3), we know that for two lines to be perpendicular $a = - b$ , which is satisfied with the given equation.
Hence, whatever be the values of $p$ and $q$ , the given pair of straight lines will always be perpendicular.
Thus, the correct option is D.

Note: Make sure to compare the given equation with the general equation of a pair of straight lines properly in order to get the values of the coefficients. Avoid any mistakes while substituting those values in the condition required for the lines to be perpendicular.