Question

What is the angle between the lines represented by the equation ${x^2} - 2pxy + {y^2} = 0$ ?
A. ${\sec ^{ - 1}}p$
B. ${\cos ^{ - 1}}p$
C. ${\tan ^{ - 1}}p$
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 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:
Given equation:
${x^2} - 2pxy + {y^2} = 0$ … (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
$h = - p$
Now, we know that the angle between two lines is calculated using:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {{p^2} - 1} }}{2}} \right|$
This on further simplification gives:
$\tan \theta = \sqrt {{p^2} - 1} $
Squaring both the sides, we get:
${\tan ^2}\theta + 1 = p^{2}$
We know that ${\tan ^2}\theta + 1 = {\sec ^2}\theta $
Therefore, $\sec \theta = p$ .
Hence,
$\theta = {\sec ^{ - 1}}p$
Thus, the correct option is A.

Note: While substituting the values in the formula, to calculate the tangent of the angle between a pair of lines, take correct values. To ensure this, compare the given equation with the general form of a pair of lines correctly and then substitute the corresponding values in the formula.