Question

What is the angle between the pair of straight lines $2{x^2} + 5xy + 2{y^2} + 3x + 3y + 1 = 0$ ?
A. \[{\cos ^{ - 1}}\left( {\dfrac{4}{5}} \right)\]
B. \[{\tan ^{ - 1}}\left( {\dfrac{4}{5}} \right)\]
C. \[\dfrac{\pi }{2}\]
D. $0$

Answer 1

Hint: A pair of straight lines are represented by a general equation of the form $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ . 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 represented by $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ 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 two straight lines:
$2{x^2} + 5xy + 2{y^2} + 3x + 3y + 1 = 0$ … (1)
General form of a pair of straight lines passing through the origin:
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ … (2)
Comparing equation (1) with the general form given in equation (2),
$a = 2$ ,
$b = 2$ ,
$c = 1$ ,
$f = \dfrac{3}{2}$ ,
$g = \dfrac{3}{2}$ and
$h = \dfrac{5}{2}$
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:
$\tan \theta = \left| {\dfrac{{2\sqrt {\dfrac{{25}}{4} - 4} }}{{2 + 2}}} \right|$
On simplifying further, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {\dfrac{9}{4}} }}{4}} \right|$
This gives us:
\[\tan \theta = \dfrac{3}{4}\]
Calculating the inverse, we get $\theta = {\tan ^{ - 1}}\left( {\dfrac{3}{4}} \right)$ .
Now, this value of ${\tan ^{ - 1}}\left( {\dfrac{3}{4}} \right)$ is equivalent to ${\cos ^{ - 1}}\left( {\dfrac{4}{5}} \right)$ .
Hence, the angle between the given pair of straight lines is ${\cos ^{ - 1}}\left( {\dfrac{4}{5}} \right)$ .
Thus, the correct option is A.

Note: Compare the given equation with the general form of a pair of straight lines to get the values and substitute those correct values in the formula to calculate the tangent of the angle between a pair of straight lines. This will avoid any further miscalculations. Students should not get confused with the values of f and g as they are not required.