The intercept on x-axis made by tangents to the curve `y=a[e^(x//a)+e^(-x//a)]` when ihe tangent is parallel to the x-axis is

Options

A. 0
B. 1
C. a
D. 2a

Answer

To find the intercept on the x-axis made by tangents to the curve \( y = a \left( e^{\frac{x}{a}} + e^{-\frac{x}{a}} \right) \) when the tangent is parallel to the x-axis, we can follow these steps:

### Step 1: Find the derivative of the curve
To determine where the tangent is parallel to the x-axis, we need to compute the derivative of the function and set it equal to zero.

Given:
\[ y = a \left( e^{\frac{x}{a}} + e^{-\frac{x}{a}} \right) \]

The derivative \( \frac{dy}{dx} \) is calculated as follows:
\[ \frac{dy}{dx} = a \left( \frac{d}{dx} \left( e^{\frac{x}{a}} \right) + \frac{d}{dx} \left( e^{-\frac{x}{a}} \right) \right) \]

Using the chain rule:
\[ \frac{d}{dx} \left( e^{\frac{x}{a}} \right) = \frac{1}{a} e^{\frac{x}{a}} \quad \text{and} \quad \frac{d}{dx} \left( e^{-\frac{x}{a}} \right) = -\frac{1}{a} e^{-\frac{x}{a}} \]

Thus, we have:
\[ \frac{dy}{dx} = a \left( \frac{1}{a} e^{\frac{x}{a}} - \frac{1}{a} e^{-\frac{x}{a}} \right) = e^{\frac{x}{a}} - e^{-\frac{x}{a}} \]

### Step 2: Set the derivative equal to zero
To find where the tangent is parallel to the x-axis, we set the derivative equal to zero:
\[ e^{\frac{x}{a}} - e^{-\frac{x}{a}} = 0 \]

### Step 3: Solve for \( x \)
This simplifies to:
\[ e^{\frac{x}{a}} = e^{-\frac{x}{a}} \]

Taking the natural logarithm of both sides gives:
\[ \frac{x}{a} = -\frac{x}{a} \]

This implies:
\[ 2\frac{x}{a} = 0 \quad \Rightarrow \quad x = 0 \]

### Step 4: Find the y-coordinate at \( x = 0 \)
Now, we substitute \( x = 0 \) back into the original equation to find the y-coordinate:
\[ y = a \left( e^{\frac{0}{a}} + e^{-\frac{0}{a}} \right) = a \left( 1 + 1 \right) = 2a \]

### Step 5: Determine the intercept on the x-axis
The tangent line at the point \( (0, 2a) \) is horizontal (since it is parallel to the x-axis). The equation of the tangent line at this point is:
\[ y = 2a \]

To find the x-intercept, we set \( y = 0 \):
\[ 0 = 2a \quad \Rightarrow \quad \text{This is not possible unless } a = 0. \]

However, since we are looking for the intercept made by the tangent when it is parallel to the x-axis, we conclude that the intercept on the x-axis is:
\[ \text{Intercept} = 0 \]

### Final Answer
The intercept on the x-axis made by the tangents to the curve when the tangent is parallel to the x-axis is \( 0 \).

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