If `x^2+y^2=(t+1/t)` and `x^4+y^4=t^2+1/t^2`, then `x^3y(dy)/(dx)=`

Options

A. 0
B. 1
C. -1
D. none of these

Answer

To solve the problem, we need to find the value of \( x^3 y \frac{dy}{dx} \) given the equations:

1. \( x^2 + y^2 = t + \frac{1}{t} \)
2. \( x^4 + y^4 = t^2 + \frac{1}{t^2} \)

### Step 1: Differentiate the first equation
We start by differentiating the first equation with respect to \( x \):

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

Using the chain rule, we get:

\[ 2x + 2y \frac{dy}{dx} = \frac{dt}{dx} - \frac{1}{t^2} \frac{dt}{dx} \]

Factoring out \( \frac{dt}{dx} \):

\[ 2x + 2y \frac{dy}{dx} = \left(1 - \frac{1}{t^2}\right) \frac{dt}{dx} \]

### Step 2: Differentiate the second equation
Next, we differentiate the second equation:

\[ \frac{d}{dx}(x^4 + y^4) = \frac{d}{dx}\left(t^2 + \frac{1}{t^2}\right) \]

Using the chain rule again, we have:

\[ 4x^3 + 4y^3 \frac{dy}{dx} = 2t \frac{dt}{dx} - \frac{2}{t^3} \frac{dt}{dx} \]

Factoring out \( \frac{dt}{dx} \):

\[ 4x^3 + 4y^3 \frac{dy}{dx} = \left(2t - \frac{2}{t^3}\right) \frac{dt}{dx} \]

### Step 3: Solve for \( \frac{dy}{dx} \)
Now we have two equations involving \( \frac{dy}{dx} \) and \( \frac{dt}{dx} \). We can express \( \frac{dt}{dx} \) from the first equation:

\[ \frac{dt}{dx} = \frac{2x + 2y \frac{dy}{dx}}{1 - \frac{1}{t^2}} \]

Substituting this into the second equation gives us a relationship between \( x, y, \frac{dy}{dx} \), and \( t \).

### Step 4: Express \( x^3 y \frac{dy}{dx} \)
We need to find \( x^3 y \frac{dy}{dx} \). From the first differentiated equation, we can isolate \( \frac{dy}{dx} \):

\[ 2y \frac{dy}{dx} = -2x + \left(1 - \frac{1}{t^2}\right) \frac{dt}{dx} \]

Now, we can multiply both sides by \( \frac{x^3 y}{2y} \):

\[ x^3 y \frac{dy}{dx} = -x^4 + \frac{x^3 y}{2y} \left(1 - \frac{1}{t^2}\right) \frac{dt}{dx} \]

### Step 5: Substitute known values
From the original equations, we know that \( x^2 + y^2 = t + \frac{1}{t} \) and \( x^4 + y^4 = t^2 + \frac{1}{t^2} \). We can use these to simplify our expressions.

After simplification, we find that:

\[ x^3 y \frac{dy}{dx} = -1 \]

Thus, the final answer is:

\[ \boxed{-1} \]