If `y = mx + 2` is parallel to a tangent to curve `e^(4y) =1 + 16x^(2)` then

Options

A. `|m| lt 1`
B. `|m| lt 1`
C. `|m| gt 1`
D. `|m| ge 1`

Answer

To solve the problem, we need to find the conditions under which the line \( y = mx + 2 \) is parallel to the tangent of the curve defined by \( e^{4y} = 1 + 16x^2 \).

### Step-by-Step Solution:

1. **Differentiate the Curve**:
We start with the equation of the curve:
\[ e^{4y} = 1 + 16x^2 \]
To find the slope of the tangent, we differentiate both sides with respect to \( x \).

Using implicit differentiation:
\[ \frac{d}{dx}(e^{4y}) = \frac{d}{dx}(1 + 16x^2) \]
The left-hand side requires the chain rule:
\[ e^{4y} \cdot \frac{d(4y)}{dx} = e^{4y} \cdot 4 \frac{dy}{dx} \]
The right-hand side differentiates to:
\[ 0 + 32x \]
Thus, we have:
\[ 4e^{4y} \frac{dy}{dx} = 32x \]

2. **Solve for \(\frac{dy}{dx}\)**:
Rearranging gives:
\[ \frac{dy}{dx} = \frac{32x}{4e^{4y}} = \frac{8x}{e^{4y}} \]

3. **Substitute \(y\) from the Line Equation**:
Since the line \( y = mx + 2 \) is parallel to the tangent, the slope \( m \) must equal \( \frac{dy}{dx} \):
\[ m = \frac{8x}{e^{4y}} \]

4. **Express \(e^{4y}\)**:
From the original curve equation, we can express \( e^{4y} \):
\[ e^{4y} = 1 + 16x^2 \]
Substitute this into the slope equation:
\[ m = \frac{8x}{1 + 16x^2} \]

5. **Analyze the Condition for Parallelism**:
For the line to be parallel to the tangent, we need to analyze the value of \( m \):
\[ |m| < 1 \]
This leads to:
\[ \left| \frac{8x}{1 + 16x^2} \right| < 1 \]

6. **Solve the Inequality**:
To solve the inequality:
\[ -1 < \frac{8x}{1 + 16x^2} < 1 \]
We can break it into two parts:
- For \( \frac{8x}{1 + 16x^2} < 1 \):
\[ 8x < 1 + 16x^2 \implies 16x^2 - 8x + 1 > 0 \]
- For \( \frac{8x}{1 + 16x^2} > -1 \):
\[ 8x > -1 - 16x^2 \implies 16x^2 + 8x + 1 > 0 \]

Both inequalities can be solved using the quadratic formula or by analyzing the discriminants.

### Final Result:
From the analysis, we find that the condition for \( |m| < 1 \) holds true under certain values of \( x \). Thus, the answer is:
\[ |m| < 1 \]