If at any point (x, y) on a curve subtangent and subnormal are of equal length, then the length of the tangent is

Options

A. `sqrt(2y)`
B. `sqrt2y`
C. y
D. none

Answer

To solve the problem, we need to find the length of the tangent when the subtangent and subnormal at any point (x, y) on a curve are of equal length.

### Step-by-Step Solution:

1. **Understanding Subtangent and Subnormal:**
- The length of the subtangent (T) at a point (x, y) on the curve is given by the formula:
\[ T = y \cdot \frac{dx}{dy} \]
- The length of the subnormal (N) at the same point is given by:
\[ N = y \cdot \frac{dy}{dx} \]

2. **Setting Up the Equation:**
- According to the problem, the subtangent and subnormal are equal:
\[ T = N \]
- Therefore, we can write:
\[ y \cdot \frac{dx}{dy} = y \cdot \frac{dy}{dx} \]

3. **Simplifying the Equation:**
- Since \(y\) is not zero, we can divide both sides by \(y\):
\[ \frac{dx}{dy} = \frac{dy}{dx} \]

4. **Cross Multiplying:**
- Cross-multiplying gives us:
\[ (dx)^2 = (dy)^2 \]
- Taking the square root of both sides results in:
\[ \frac{dy}{dx} = \pm 1 \]

5. **Finding the Length of the Tangent:**
- The length of the tangent (L) at the point (x, y) is given by:
\[ L = y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
- Substituting \(\frac{dy}{dx} = \pm 1\) into the formula:
\[ L = y \sqrt{1 + 1^2} = y \sqrt{2} \]

6. **Final Result:**
- Thus, the length of the tangent is:
\[ L = \sqrt{2} \cdot y \]

### Conclusion:
The length of the tangent when the subtangent and subnormal are equal is \( \sqrt{2} \cdot y \).