Question

If the equation ${x^2} + {k_1}{y^2} + {k_2}xy = 0$ represents a pair of perpendicular lines, then which of the following options is correct?
A. ${k_1} = - 1$
B. ${k_1} = 2{k_2}$
C. $2{k_1} = {k_2}$
D. None of these

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:
General equation of the pair of lines passing through the origin:
$a{x^2} + 2hxy + b{y^2} = 0$ … (1)
Given equation of a pair of perpendicular lines:
${x^2} + {k_1}{y^2} + {k_2}xy = 0$
Comparing this equation with the general form provided in equation (1), we get:
$a = 1,b = {k_1}$ and $h = \dfrac{{{k_2}}}{2}$ … (2)
Now, we know that the tangent of the angle between a pair of lines is given by:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
As the given pair of lines is perpendicular, therefore $\theta = \dfrac{\pi }{2}$ .
Substituting this value and all the other values from (2), we get:
$\tan \dfrac{\pi }{2} = \left| {\dfrac{{2\sqrt {{{\dfrac{{{k_2}}}{4}}^2} - {k_1}} }}{{1 + {k_1}}}} \right|$
We know that the value of $\tan \dfrac{\pi }{2}$ is undefined (or infinity), hence,
$1 + {k_1} = 0$
This gives ${k_1} = - 1$ .
Hence, if ${x^2} + {k_1}{y^2} + {k_2}xy = 0$ represents a pair of perpendicular lines, then ${k_1} = - 1$ .
Thus, the correct option is A.

Note: While finding out the condition required for the given pair of lines to be perpendicular to each other, keep in mind to substitute the correct values in the formula for tangent of the angle between a pair of lines. This will avoid any further miscalculations. If students remember the condition of perpendicular lines then it is easy to answer such questions.