NIMCET 2010 — Mathematics PYQ

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Question

NIMCET 2010 — Mathematics PYQ

NIMCET | Mathematics | 2010

The angle of intersection of the cardioids $r = a(1 + \cos \theta)$ and $r = a(1 - \cos \theta)$ is:

Choose the correct answer:

Explanation

Step 1: Angle for the first cardioid ( $r_1 = a(1 + \cos \theta)$ )

Differentiating with respect to $\theta$ :

\frac{dr_1}{d\theta} = -a \sin \theta

We use the formula $\tan \phi = \frac{r}{dr/d\theta}$ :

\tan \phi_1 = \frac{a(1 + \cos \theta)}{-a \sin \theta} = \frac{2 \cos^2 (\theta/2)}{-2 \sin (\theta/2) \cos (\theta/2)}

\tan \phi_1 = -\cot(\theta/2) = \tan(\pi/2 + \theta/2)

\phi_1 = \frac{\pi}{2} + \frac{\theta}{2}

Step 2: Angle for the second cardioid ( $r_2 = a(1 - \cos \theta)$ )

Differentiating with respect to $\theta$ :

\frac{dr_2}{d\theta} = a \sin \theta

\tan \phi_2 = \frac{a(1 - \cos \theta)}{a \sin \theta} = \frac{2 \sin^2 (\theta/2)}{2 \sin (\theta/2) \cos (\theta/2)}

\tan \phi_2 = \tan(\theta/2)

\phi_2 = \frac{\theta}{2}

Step 3: Calculate the angle of intersection

The angle of intersection $\alpha$ is given by:

\alpha = |\phi_1 - \phi_2|

\alpha = \left| \left( \frac{\pi}{2} + \frac{\theta}{2} \right) - \left( \frac{\theta}{2} \right) \right|

\alpha = \frac{\pi}{2}

Conclusion:

The two cardioids intersect orthogonally (at a right angle).

Correct Option:

(a) $\pi/2$