Question

Find \(\frac{dy}{dx}\) if x = a (θ + sin θ) and y = a (1 - cos θ)
1. \(tan \ \frac{\theta}{2}\)
2. tan θ
3. sin θ
4. None of these

Answer 1

Correct Answer - Option 1 : \(tan \ \frac{\theta}{2}\)

CONCEPT:

• \(\frac{{d\left( {\sin x} \right)}}{{dx}} = \cos x\)
• \(\frac{{d\left( {\cos x} \right)}}{{dx}} = \; - \sin x\)

If x = f(t), y = g(t), where t is a parameter, then \(\frac{{dy}}{{dx}} = \frac{{g'\left( t \right)}}{{f'\left( t \right)}}\)

CALCULATION:

Given: x = a (θ + sin θ) and y = a (1 - cos θ)

Here, we have to find \(\frac{dy}{dx}\)

So, first we have to find dx/dθ and dy/dθ

⇒ \(\frac{dx}{dθ } = a + a \ cos \ θ \)

⇒ \(\frac{dy}{dθ } = a \ sin \ θ \)

As we know that, if x = f(t), y = g(t), where t is a parameter, then \(\frac{{dy}}{{dx}} = \frac{{g'\left( t \right)}}{{f'\left( t \right)}}\)

⇒ \(\frac{dy}{dx} = \frac{a \ sin \ θ}{a(1 + cos \ θ)} = tan \ \frac{θ}{2}\)

Hence, correct option is 1.