The x intercept of the tangent to a curve f(x,y) = 0 is equal to the ordinate of the point of contact. Then the value of `(d^(2)x)/(dy^(2))` at the point (1,1) on the curve is `"_____".`

Answer

To solve the problem, we need to find the value of \(\frac{d^2x}{dy^2}\) at the point (1,1) on the curve defined by \(f(x,y) = 0\), given that the x-intercept of the tangent to the curve at a point is equal to the ordinate (y-coordinate) of that point.

### Step-by-Step Solution:

1. **Identify the Point of Tangency**:
Let the point of tangency on the curve be \((x_1, y_1)\). We need to find the tangent line at this point.

2. **Find the X-Intercept of the Tangent**:
The x-intercept of the tangent line can be found by setting \(y = 0\) in the equation of the tangent. According to the problem, this x-intercept is equal to the ordinate \(y_1\). Therefore, the x-intercept is \((y_1, 0)\).

3. **Determine the Slope of the Tangent**:
The slope \(m\) of the tangent line at the point \((x_1, y_1)\) can be expressed using the two-point form of the slope:
\[ m = \frac{y_1 - 0}{x_1 - y_1} = \frac{y_1}{x_1 - y_1} \]
However, we can also express the slope in terms of derivatives:
\[ \frac{dy}{dx} = \frac{y}{x - y} \]

4. **Reciprocal to Find \(\frac{dx}{dy}\)**:
We need to find \(\frac{dx}{dy}\), which is the reciprocal of \(\frac{dy}{dx}\):
\[ \frac{dx}{dy} = \frac{x - y}{y} \]

5. **Consider Two Cases**:
From the expression for \(\frac{dx}{dy}\), we can consider two cases:
- Case 1: \(\frac{dx}{dy} = \frac{x + y}{y}\)
- Case 2: \(\frac{dx}{dy} = \frac{x - y}{y}\)

6. **Differentiate to Find \(\frac{d^2x}{dy^2}\)**:
Now, we differentiate \(\frac{dx}{dy}\) with respect to \(y\) to find \(\frac{d^2x}{dy^2}\):
- For Case 1:
\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\frac{x + y}{y}\right) = \frac{1}{y} \cdot \frac{dx}{dy} - \frac{(x + y)}{y^2} \]
- For Case 2:
\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\frac{x - y}{y}\right) = \frac{1}{y} \cdot \frac{dx}{dy} - \frac{(x - y)}{y^2} \]

7. **Evaluate at the Point (1,1)**:
Substitute \(x = 1\) and \(y = 1\) into both cases:
- For both cases, we find that:
\[ \frac{d^2x}{dy^2} = -1 \]

### Final Answer:
Thus, the value of \(\frac{d^2x}{dy^2}\) at the point (1,1) is \(-1\).