Question

What is the condition for two lines represented by the equation $a{x^2} + 2hxy + b{y^2} = 0$ to be perpendicular?
A. $ab = - 1$
B. $a = - b$
C. $a = b$
D. $ab = 1$

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.

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:
Equation of the pair of lines:
$a{x^2} + 2hxy + b{y^2} = 0$
Now, we know that the tangent of the angle between a pair of lines is given by:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
For a pair of lines to be perpendicular, $\theta = \dfrac{\pi }{2}$ .
Substituting this value, we get:
$\tan \dfrac{\pi }{2} = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
We know that the value of $\tan \dfrac{\pi }{2}$ is undefined (or infinity), hence,
$a + b = 0$
This gives $a = - b$ .
Hence, for a pair of straight lines, of the form $a{x^2} + 2hxy + b{y^2} = 0$ , to be perpendicular, $a = - b$ .
Thus, the correct option is B.

Note: While calculating the condition required for the given equation of a pair of lines to be perpendicular, just keep in mind to substitute the correct values in the formula. This will avoid any further mistakes.