Question

Angle between $x=2$ and $x-3 y=6$ is

A) $\infty$

B) $\tan ^{-1}(3)$

C) $\tan ^{-1}\left(\dfrac{1}{3}\right)$

D) None of the above

Answer 1

Hint: The formula to find angle between two lines is given by $\tan \theta=\left|\dfrac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|$ and after using $\tan ^{-1}$ on both the sides of the equation will imply $\tan ^{-1}(\tan \theta)=\tan ^{-1}\left|\dfrac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|$. Thus the direct formula is given by $\theta=\tan ^{-1}\left|\dfrac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|$. Here $m_{1}$ and $m_{2}$ are slopes of their respective lines.

Complete step-by-step solution:

If the angle between the lines is zero then the lines are parallel. If it is 90 degrees the lines are perpendicular. The terms $m_{1}$ and $m_{2}$ are slopes of equation of lines.

Now we consider general equation for a straight line $y=m x+c$ and consider the given equation $x=2$ and comparing $x=2$ with the general equation $y=m_{1} x+c$ we get no slope. This is due to the fact that there is no y term in this equation since, $x=2$ is a line which is perpendicular to the $x$ -axis.

Now, we will find the slope for the equation $x-3 y=6$. Now put the required terms to the right side of the sign and we get $3 y=x-6$ resulting into $y=\dfrac{x}{3}-2$ and $y=\dfrac{x}{3}-2$ with standard equation of line $y=m_{2} x+c$ we get $m_{2}=\dfrac{1}{3}$. As we know that the slope of the angle is made with the $x$ axis therefore, the diagram for the two equations along with the slope is given below.

As we know that the slope is given by the formula $\tan (\theta)=m_{2}$ therefore, we get $\tan (\theta)=\dfrac{1}{3}$. Now we will take tangent operation to the right side of the equation. Thus, we get $\theta=\tan ^{-1}\left(\dfrac{1}{3}\right)$. Now, if we notice here we see that this angle is actually the angle between the line $3 \mathrm{y}=\mathrm{x}-6$ and the $\mathrm{x}$ -axis. But we need to find the angle between the lines $x=2$ and $3 y=x-6 .$ So, we will use the concept of a triangle which states that the sum of all interior angles of a triangle is 180 degrees. 

Therefore, we get

$90^{\circ}+\theta+x=180^{\circ}$

$\Rightarrow x=180^{\circ}-90^{\circ}-\theta$

As we know that $\theta=\tan ^{-1}\left(\dfrac{1}{3}\right)$ therefore, we get $x=180^{\circ}-90^{\circ}-\tan ^{-1}\left(\dfrac{1}{3}\right)$

$\Rightarrow x=90^{\circ}-\tan ^{-1}\left(\dfrac{1}{3}\right)$

Since, $90^{\circ}=\dfrac{\pi}{2}$ therefore, we now have $x=\dfrac{\pi}{2}-\tan ^{-1}\left(\dfrac{1}{3}\right)$. Now we will here use a trigonometry formula which is given by $\cot ^{-1}(\theta)+\tan ^{-1}(\theta)=\dfrac{\pi}{2}$ or $\cot ^{-1}(\theta)=\dfrac{\pi}{2}-\tan ^{-1}(\theta)$ in the equation. Thus, after this we get $x=\cot ^{-1}\left(\dfrac{1}{3}\right)$. By taking cotangent to the right side of the equation we will get $\cot x=\dfrac{1}{3}$. Now as we know that $\cot x=\dfrac{1}{\tan x}$ therefore we get $\dfrac{1}{\tan x}=\dfrac{1}{3}$. This results in $\tan x=3$. By taking the tangent to the right side of the equation we will get $x=\tan ^{-1}(3)$.

Therefore, the angle between the lines $x=2$ and $3 y=x-6($ or $x-3 y=6)$ is $x=\tan^{-1}(3) .$ 

Hence, the correct option is (B).

Note: Notice while placing the trigonometric terms to the either side of the equation. This means that if we are taking inverse terms to the either side of an equal sign then it will turn into a simple operation which is without inverse operation and vice-versa. As we are here given to find the angle between the two lines one can make mistakes while drawing the angles. By watching the diagram we can clear the confusion. Just need to remember here that since, there is a topic about slope then it will be made with the $x$ axis here. And then we will find the angle between the given lines. If this point is not maintained then the answer will not be correct at all. One can be confused by reading the line "to find the angle between the lines" and use the formula given by $\tan \theta=\left|\dfrac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|$ We are not given the second slope here as the equation of line is $x=2 .$ Therefore, this formula is not valid here.