Question

What is the angle between the pair of lines given by the equation ${x^2} + 2xy - {y^2} = 0$ ?
A. $\dfrac{\pi }{3}$
B. $\dfrac{\pi }{6}$
C. $\dfrac{\pi }{2}$
D. $0$

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 $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$. Using this formula we will calculate the angle between both the lines.

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 of a pair of straight lines:
${x^2} + 2xy - {y^2} = 0$ … (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),
$a = 1$ ,
$b = - 1$ and
$h = 1$
Now, we know that the tangent of the angle, let’s say $\theta $ , between two lines is calculated using:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values of the variables, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {1 - ( - 1)} }}{{1 + ( - 1)}}} \right|$
On simplifying further, we get:
$\tan \theta = \infty $
Calculating the inverse, we get $\theta = \dfrac{\pi }{2}$ .
Hence, the angle between the given lines is ${90^ \circ }$ .
Thus, the correct option is C.

Note: Compare the coefficients of the given equation to the general equation of a pair of straight lines properly. Then, substitute the corresponding values in the formula to calculate the tangent of the angle between the lines. Make sure to compare and substitute the values correctly to avoid any mistakes. We can directly find the angle between two lines using condition of perpendicular lines.