Consider the curve `x=a(cos theta+thetasintheta)` and `y=a(sin theta-thetacostheta)`.
what is `(d^(2)y)/(dx^(2))` equal to ?

Options

A. `sec^(2)theta`
B. `-"cosec"^(2)theta`
C. `(sec^(2)theta)/(atheta)`
D. None of these

Answer

To find \(\frac{d^2y}{dx^2}\) for the given parametric equations \(x = a(\cos \theta + \theta \sin \theta)\) and \(y = a(\sin \theta - \theta \cos \theta)\), we will follow these steps:

### Step 1: Find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\)

1. Differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = a\left(\cos \theta - \theta \sin \theta - \sin \theta\right) = a\left(\cos \theta - \sin \theta - \theta \sin \theta\right) \]

2. Differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = a\left(-\sin \theta + \theta \cos \theta + \cos \theta\right) = a\left(\cos \theta - \sin \theta + \theta \cos \theta\right) \]