NIMCET 2021 — Mathematics PYQ

Step 1: Factorize the given equation.

The equation of the pair of lines passing through the origin is:

To find the individual lines, we factorize the quadratic expression:

So, the two individual lines are:

1. $L_1:x-4y=0⟹y=\frac{1}{4}x(\text{slope }{m}_1=\frac{1}{4})$

2. $L_2:x-3y=0⟹y=\frac{1}{3}x(\text{slope }{m}_2=\frac{1}{3})$

Step 2: Find $\tan \theta$ between the lines.

The formula for the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is:

Substituting the values:

Step 3: Simplify the required expression.

We need to find the value of:

Divide both the numerator and the denominator by $\cos \theta$:

Step 4: Substitute the value of $\tan \theta$.

Final Answer: