Question

If the angle between two lines represented by $2{x^2} + 5xy + 3{y^2} + 7y + 4 = 0$ is ${\tan ^{ - 1}}m$ , then what is the value of $m$ ?
A. $\dfrac{1}{5}$
B. $1$
C. $\dfrac{7}{5}$
D. $7$

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

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 a pair of straight lines:
$2{x^2} + 5xy + 3{y^2} + 7y + 4 = 0$ … (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),
$a = 2$ ,
$b = 3$ ,
$c = 4$ ,
$f = \dfrac{7}{2}$ ,
$g = 0$ 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:
$\tan \theta = \left| {\dfrac{{2\sqrt {\dfrac{{25}}{4} - 6} }}{{2 + 3}}} \right|$
On simplifying further, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {\dfrac{1}{4}} }}{5}} \right|$
This gives us:
$\tan \theta = \dfrac{1}{5}$
Calculating the inverse, we get $\theta = {\tan ^{ - 1}}\left( {\dfrac{1}{5}} \right)$ .
It is also given that the angle between the two lines is ${\tan ^{ - 1}}m$ .
Hence, on comparing, $m = \dfrac{1}{5}$ .
Thus, the correct option is A.

Note: Make sure to substitute the correct values in the formula to calculate the tangent of the angle between the lines. This can be ensured by comparing the given equation to the general form of a pair of straight lines and getting the corresponding values of the coefficients. Students should not get confused with the values of f and g, as we need to neglect while finding the angle.