🤖 AI Generated Solution:

**Correct Option if MCQ:** [Not provided, solution follows general format] **Reasoning:** * \(\frac{dy}{dx} = -\frac{b}{a} \tan \theta\) * \(\frac{d^{2}y}{dx^{2}} = \frac{2b}{a^{2}} \sec^{4}\theta \csc\theta\) * At \(\theta=\frac{\pi}{4}\), \(\frac{d^{2}y}{dx^{2}} = \frac{4b}{a^{2}}\)

🤖 AI Generated Solution:

**Correct Option if MCQ:** Not Applicable **Reasoning:** * \(\frac{dx}{d\theta} = a \left(-\sin \theta + \frac{1}{\tan \frac{\theta}{2}} \sec^2 \frac{\theta}{2} \cdot \frac{1}{2}\right) = a(-\sin \theta + \frac{1}{\sin \frac{\theta}{2} \cos \frac{\theta}{2}}) = a(-\sin \theta + \frac{2}{\sin \theta}) = a \frac{2 - \sin^2 \theta}{\sin \theta}\) * \(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{\cos \theta}{a \frac{2 - \sin^2 \theta}{\sin \theta}} = \frac{\sin \theta \cos \theta}{a (2 - \sin^2 \theta)}\) * \(\frac{d^2y}{dx^2} = \frac{d}{d\theta}\left(\frac{dy}{dx}\right) \frac{d\theta}{dx} = \frac{a(2-3\sin^2\theta)\cos\theta}{(a(2-\sin^2\theta))^2} \cdot \frac{a(2-\sin^2\theta)}{\sin\theta(2-\sin^2\theta)} = \frac{a(2-3\sin^2\theta)\cos\theta}{a^2\sin\theta(2-\sin^2\theta)^2}\). At \(\theta = \frac{\pi}{4}\), \(\sin\theta = \cos\theta = \frac{1}{\sqrt{2}}\). \(\frac{d^2y}{dx^2} = \frac{a(2-3/2)(1/\sqrt{2})}{a^2(1/\sqrt{2})(2-1/2)^2} = \frac{a(1/2)(1/\sqrt{2})}{a^2(1/\sqrt{2})(3/2)^2} = \frac{a/2}{a^2(9/4)} = \frac{1}{a(9/2)} = \frac{2}{9a}\).