Question

The angle between the lines $3x - y + 5 = 0$ , $x + 3y - 2 = 0$ is
A. $\dfrac{\pi }{2}$
B. $\dfrac{\pi }{4}$
C. $0$
D. $\dfrac{\pi }{6}$

Answer 1

Hint : Linear equations are defined for the lines of the coordinate system and represent straight lines. These equations are of the first order. The slope of a line tells the steepness of a line, it can be defined as a change in y per unit change in x .To find the angle between two lines we have to first find their slopes.

Complete step-by-step answer :
Equation of a line is - $y = mx + c$
where $m$ is the slope of the line and $c$ is the intercept of the line on the y-axis.
So we first concert the given lines to this form,
$3x - y + 5 = 0$ can be rearranged as $y = 3x + 5$
The slope of line 1 is, ${m_1} = 3$
$x + 3y - 2 = 0$ can be rewritten as -
$3y = - x + 2 \\\ \Rightarrow y = \dfrac{{ - 1}}{3}x + \dfrac{2}{3} \;$
The slope of the second line is, ${m_2} = \dfrac{{ - 1}}{3}$
Now the angle between two lines can be found out by the formula –
$\Rightarrow \tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|\\\ = \left| {\dfrac{{3 - (\dfrac{{ - 1}}{3})}}{{1 + 3 \times (\dfrac{{ - 1}}{3})}}} \right|\\\ = \left| {\dfrac{{3 + \dfrac{1}{3}}}{{1 - 1}}} \right| = \left| {\dfrac{{\dfrac{{10}}{3}}}{0}} \right| \\\ \Rightarrow \tan \theta = \infty \\\ \therefore \theta = \dfrac{\pi }{2} \;$
Thus, the two lines are perpendicular to each other . The angle between them is $\dfrac{\pi }{2}$ .
So, the correct answer is “Option A”.

Note : The intersection of two lines forms an angle between them. The value of the angle is found out using the slopes of the lines, slope is also called a tangent. To find the slope of a line we first convert it into the slope-intercept form and then put the value of slopes in the suitable formula. The intercept on the y-axis is the distance between the origin and the point at which the line cuts the y-axis.