Question

Which one of the following equations represent a pair of straight lines perpendicular to each other?
A. $2{x^2} = 2y(2x + y)$
B. ${x^2} + {y^2} + 3 = 0$
C. $2{x^2} = y(2x + y)$
D. ${x^2} = 2(x - y)$

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 the straight lines 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, $a = - b$ .
Now, we’ll simplify each of the equations in the provided options and check them.
For option A:
$2{x^2} - 4xy - 2{y^2} = 0$
Comparing this equation with equation (1), we get:
$a = 2,b = - 2$
That means $a = - b$
From equation (3), we know that for two lines to be perpendicular $a = - b$ , which is satisfied with the equation given in option A.
While for equations given in other options, this condition was not satisfied.
Thus, the correct option is A.

Note: In the above question, once the condition for two straight lines to be perpendicular to each other is figured out, the question can be solved easily. Thus, it will be convenient for a student to understand and learn the conditions required for two lines to be perpendicular, parallel or coincident.