A curve is represented parametrically by the equations `x = e^t cos t and y = e^t sin t` where t is a parameter. Then The relation between the parameter 't' and the angle a between the tangent to the given curve andthe x-axis is given by, 't' equals

Options

A. `(pi)/(2)-alpha`
B. `(pi)/(4)+alpha`
C. `alpha-(pi)/(4)`
D. `(pi)/(4)-alpha`

Answer

To find the relation between the parameter \( t \) and the angle \( \alpha \) between the tangent to the given curve and the x-axis, we start with the parametric equations:

\[ x = e^t \cos t \]
\[ y = e^t \sin t \]

### Step 1: Differentiate \( x \) and \( y \) with respect to \( t \)

We need to find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).

For \( x \):
\[ \frac{dx}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t = e^t (\cos t - \sin t) \]

For \( y \):
\[ \frac{dy}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t = e^t (\sin t + \cos t) \]

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

Using the chain rule, we can find \( \frac{dy}{dx} \) as follows:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{e^t (\sin t + \cos t)}{e^t (\cos t - \sin t)} \]

The \( e^t \) terms cancel out:
\[ \frac{dy}{dx} = \frac{\sin t + \cos t}{\cos t - \sin t} \]

### Step 3: Relate \( \frac{dy}{dx} \) to the angle \( \alpha \)

The slope of the tangent line to the curve is equal to \( \tan \alpha \):
\[ \tan \alpha = \frac{dy}{dx} = \frac{\sin t + \cos t}{\cos t - \sin t} \]

### Step 4: Simplify \( \tan \alpha \)

We can express \( \tan \alpha \) in terms of \( t \):
\[ \tan \alpha = \frac{\sin t + \cos t}{\cos t - \sin t} \]

### Step 5: Use the tangent addition formula

We can express \( \tan \alpha \) using the tangent addition formula:
\[ \tan \alpha = \frac{1 + \tan t}{1 - \tan t} \]

This implies:
\[ \alpha = \tan^{-1}\left(\frac{1 + \tan t}{1 - \tan t}\right) \]

### Step 6: Solve for \( t \)

From the above equation, we can derive:
\[ \tan \alpha = \tan\left(\frac{\pi}{4} + t\right) \]

Thus, we can equate:
\[ \alpha = \frac{\pi}{4} + t \]

Rearranging gives:
\[ t = \alpha - \frac{\pi}{4} \]

### Final Answer

The relation between the parameter \( t \) and the angle \( \alpha \) is:
\[ t = \alpha - \frac{\pi}{4} \]