The tangent to a given curve is perpendicular to x-axis if

Options

A. `(dy)/(dx)=0`
B. `(dy)/(dx)=1`
C. `(dx)/(dy)=0`
D. `(dx)/(dy)=1`

Answer

To determine the condition under which the tangent to a given curve is perpendicular to the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Tangents and Slopes**:
- The slope of the x-axis is 0 (since it is a horizontal line).
- For a tangent to be perpendicular to the x-axis, its slope must be undefined or infinite.

2. **Identify the Slope of the Tangent**:
- The slope of the tangent to a curve at a point is given by the derivative \( \frac{dy}{dx} \).

3. **Set Up the Condition for Perpendicularity**:
- For two lines to be perpendicular, the product of their slopes must equal -1.
- Let \( m_1 \) be the slope of the x-axis (which is 0), and \( m_2 \) be the slope of the tangent \( \frac{dy}{dx} \).
- The condition for perpendicularity can be expressed as:
\[ m_1 \cdot m_2 = -1 \]
- Substituting \( m_1 = 0 \):
\[ 0 \cdot m_2 = -1 \]
- This equation does not hold true for any finite value of \( m_2 \).

4. **Conclusion about the Slope**:
- Since the product cannot equal -1, we conclude that for the tangent to be perpendicular to the x-axis, the slope \( \frac{dy}{dx} \) must be undefined or infinite.
- This occurs when:
\[ \frac{dx}{dy} = 0 \]
- This indicates that the change in \( x \) with respect to \( y \) is zero, meaning that the tangent line is vertical.

### Final Condition:
Thus, the condition for the tangent to a given curve to be perpendicular to the x-axis is:
\[ \frac{dx}{dy} = 0 \]