If `x(t)=2sqrt2 cost sqrt(sin 2t) and y(t)=2sqrt2 sint sqrt(sin 2t)`, where `t in (o,pi/2)` then the value of `{(1+(dy/dx)^2)/((d^2 y)/dx^2)}` at `x=pi/4`

Options

A. `-1/2`
B. `1/2`
C. 2
D. `-2/3`

Answer

To solve the problem, we need to find the value of \(\frac{1 + \left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}\) at \(x = \frac{\pi}{4}\) given the parametric equations:

\[ x(t) = 2\sqrt{2} \cos(t) \sqrt{\sin(2t)} \]
\[ y(t) = 2\sqrt{2} \sin(t) \sqrt{\sin(2t)} \]

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

First, we differentiate \(x(t)\) and \(y(t)\) with respect to \(t\).

\[ \frac{dx}{dt} = \frac{d}{dt} \left( 2\sqrt{2} \cos(t) \sqrt{\sin(2t)} \right) \]

Using the product rule:

\[ \frac{dx}{dt} = 2\sqrt{2} \left( -\sin(t) \sqrt{\sin(2t)} + \cos(t) \cdot \frac{1}{2\sqrt{\sin(2t)}} \cdot 2\cos(2t) \right) \]

This simplifies to:

\[ \frac{dx}{dt} = 2\sqrt{2} \left( -\sin(t) \sqrt{\sin(2t)} + \frac{\cos(t) \cos(2t)}{\sqrt{\sin(2t)}} \right) \]

Now for \(y(t)\):

\[ \frac{dy}{dt} = \frac{d}{dt} \left( 2\sqrt{2} \sin(t) \sqrt{\sin(2t)} \right) \]

Using the product rule again:

\[ \frac{dy}{dt} = 2\sqrt{2} \left( \cos(t) \sqrt{\sin(2t)} + \sin(t) \cdot \frac{1}{2\sqrt{\sin(2t)}} \cdot 2\cos(2t) \right) \]

This simplifies to:

\[ \frac{dy}{dt} = 2\sqrt{2} \left( \cos(t) \sqrt{\sin(2t)} + \frac{\sin(t) \cos(2t)}{\sqrt{\sin(2t)}} \right) \]

### Step 2: Find \(\frac{dy}{dx}\)

Using the chain rule:

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

Substituting the derivatives we found:

\[ \frac{dy}{dx} = \frac{2\sqrt{2} \left( \cos(t) \sqrt{\sin(2t)} + \frac{\sin(t) \cos(2t)}{\sqrt{\sin(2t)}} \right)}{2\sqrt{2} \left( -\sin(t) \sqrt{\sin(2t)} + \frac{\cos(t) \cos(2t)}{\sqrt{\sin(2t)}} \right)} \]

### Step 3: Find \(\frac{d^2y}{dx^2}\)

Using the formula:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} \]

This requires calculating \(\frac{d}{dt}\left(\frac{dy}{dx}\right)\) and \(\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}\).

### Step 4: Evaluate at \(t = \frac{\pi}{4}\)

At \(t = \frac{\pi}{4}\):

- Calculate \(x\) and \(y\) to confirm \(x = \frac{\pi}{4}\).
- Substitute \(t = \frac{\pi}{4}\) into the expressions for \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) to find their values.

### Step 5: Substitute into the expression

Finally, substitute the values of \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) into the expression:

\[ \frac{1 + \left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}} \]