Question

Find the angle between the lines y - 5x + 2 = 0 and x = y + 9 . 
1. ​\(\rm \tan^{-1} \left(\frac 1 4 \right)\)
2. \(\rm \tan^{-1} \left(\frac 1 3 \right)\)
3. \(\rm \tan^{-1} \left(\frac 2 3 \right)\)
4. \(\rm \tan^{-1} \left(\frac 1 2 \right)\)

Answer 1

Correct Answer - Option 3 : \(\rm \tan^{-1} \left(\frac 2 3 \right)\)

Concept:

The angle between the lines y = m1x + c1 and y = m2x + c2 is given by  tan θ = \(\rm \left|\frac{m_1 - m_2}{1+m_1m_2} \right |\)

Calculation:

We have given equation of lines are,  y - 5x + 2 = 0 and x = y + 9 

⇒ y = 5x - 2 

⇒ m1 = 5            ____( i ) 

and , y = x - 9  

⇒ m2 = 1           ____( ii ) 

We know that , \(\rm tanθ = \left | \frac{m_{1}-m_{2}}{1+m_{1}m_{2}} \right |\) 

⇒ tan θ = \(\rm\left | \frac{5-1}{1+5} \right |\) = 4/6 = 2/3

∴ θ = \(\rm \tan^{-1} \left(\frac 2 3 \right)\)

The correct option is 3.