NIMCET 2008 — Mathematics PYQ

1. Identify the General Equation

A pair of straight lines passing through the origin is represented by:

Comparing this with the given equation:

• $a={\tan }^2\theta +{\cos }^2\theta$

• $h=\tan \theta$

• $b=-{\sin }^2\theta$

2. Equation of Angle Bisectors

The joint equation of the bisectors of the angles between the lines $a{x}^2+2hxy+b{y}^2=0$ is given by:

3. Substitute the Values

Let's calculate $a-b$ and $h$ using $\theta =\frac{\pi }{3}$:

• $\tan \frac{\pi }{3}=\sqrt{3}⟹h=\sqrt{3}$

• $\cos \frac{\pi }{3}=\frac{1}{2}⟹{\cos }^2\theta =\frac{1}{4}$

• $\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}⟹{\sin }^2\theta =\frac{3}{4}$

Now, find $a-b$:

Since ${\cos }^2\theta +{\sin }^2\theta =1$:

4. Form the Bisector Equation

Substitute $a-b=4$ and $h=\sqrt{3}$ into the bisector formula:

5. Use the Given Bisector $y=mx$

Since $y=mx$ is a bisector, it must satisfy the joint equation of the bisectors. Substitute $y=mx$ into the equation:

Dividing by $x^2$ (since $x\ne 0$):

Rearranging the terms:

Conclusion

The value of $\sqrt{3}{m}^2+4m$ is $\sqrt{3}$.

Correct Option: (c)