If `h` denotes the A.M. and k denote G.M. of t e intercept made on axes by the lines passing through `(1, 1)` then `(h,k)` lies on

Options

A. `y^(2)=2x`
B. `y^(2)=4x`
C. `y = 2x`
D. `x +y = 2xy`

Explanation

To solve the problem, we need to find the relationship between the arithmetic mean (A.M.) and geometric mean (G.M.) of the intercepts made on the axes by the lines passing through the point (1, 1).

### Step-by-Step Solution:

1. **Define the Intercepts**:
Let the x-intercept of the line be \( A \) and the y-intercept be \( B \).

2. **Calculate A.M. and G.M.**:
- The arithmetic mean \( h \) of the intercepts \( A \) and \( B \) is given by:
\[ h = \frac{A + B}{2} \]
- The geometric mean \( k \) of the intercepts \( A \) and \( B \) is given by:
\[ k = \sqrt{AB} \]

3. **Equation of the Line**:
The equation of the line in intercept form is:
\[ \frac{x}{A} + \frac{y}{B} = 1 \]
Since the line passes through the point (1, 1), we can substitute \( x = 1 \) and \( y = 1 \):
\[ \frac{1}{A} + \frac{1}{B} = 1 \]

4. **Simplifying the Equation**:
Multiplying through by \( AB \) gives:
\[ B + A = AB \]
Rearranging this, we have:
\[ AB - A - B = 0 \]

5. **Substituting A.M. and G.M.**:
From the definitions of \( h \) and \( k \):
- \( A + B = 2h \)
- \( AB = k^2 \)

Substituting these into the equation \( AB - A - B = 0 \):
\[ k^2 - 2h = 0 \]
This can be rewritten as:
\[ k^2 = 2h \]

6. **Relating \( h \) and \( k \)**:
Since \( h \) and \( k \) correspond to \( x \) and \( y \) respectively, we can write:
\[ y^2 = 2x \]

7. **Conclusion**:
Therefore, the pair \( (h, k) \) lies on the curve described by the equation:
\[ y^2 = 2x \]

### Final Answer:
The correct option is **Option 1: \( y^2 = 2x \)**.