Question

If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2y}{dx^2}=-1$ at $t=K, K>0$, then $\lim_{t \to K} \frac{y}{x} =$

• A. $\frac{2}{\pi}$
• B. $\frac{\pi-2}{2}$
• C. $\frac{2}{\pi-2}$
• D. $\frac{\pi}{2}$

Hint:

Remember the formula for the second derivative of a parametric equation: $\frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt}$. A common mistake is forgetting to divide by $dx/dt$. Using trigonometric half-angle identities often simplifies the differentiation process.

Answer 1

The Correct Option is C

Step 1: Calculate the first derivative, $\frac{dy}{dx}$}.  
First, find the derivatives with respect to the parameter $t$. \[ \frac{dx}{dt} = 1 - \cos t, \frac{dy}{dt} = \sin t. \] The first derivative is given by the chain rule: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\sin t}{1-\cos t}. \] Using half-angle identities, this simplifies to $\frac{2\sin(t/2)\cos(t/2)}{2\sin^2(t/2)} = \cot(t/2)$. 

Step 2: Calculate the second derivative, $\frac{d^2y}{dx^2}$.} 
We use the formula $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) / \frac{dx}{dt}$. \[ \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}(\cot(t/2)) = -\csc^2(t/2) \cdot \frac{1}{2}. \] \[ \frac{d^2y}{dx^2} = \frac{-\frac{1}{2}\csc^2(t/2)}{1-\cos t} = \frac{-\frac{1}{2}\csc^2(t/2)}{2\sin^2(t/2)} = -\frac{1}{4\sin^4(t/2)} = -\frac{1}{4}\csc^4(t/2). \] 

Step 3: Use the given condition to find the value of K. 
We are given that at $t=K$, $\frac{d^2y}{dx^2} = -1$. \[ -\frac{1}{4}\csc^4(K/2) = -1 \implies \csc^4(K/2) = 4. \] Taking the fourth root, $\csc(K/2) = \sqrt{2}$ (since cosecant of half angle for cycloid is positive). \[ \sin(K/2) = \frac{1}{\sqrt{2}}. \] Since $K>0$, the smallest positive solution is $K/2 = \pi/4$, which gives $K = \pi/2$. 

Step 4: Evaluate the required limit. 
We need to find $\lim_{t \to K} \frac{y}{x} = \lim_{t \to \pi/2} \frac{1-\cos t}{t-\sin t}$. Since the function is continuous at $t=\pi/2$, we can directly substitute the value. \[ \frac{1-\cos(\pi/2)}{\pi/2 - \sin(\pi/2)} = \frac{1-0}{\pi/2 - 1} = \frac{1}{(\pi-2)/2}. \] \[ \boxed{\frac{2}{\pi-2}}. \]