A normal at `P(x , y)` on a curve meets the x-axis at `Q` and `N` is the foot of the ordinate at `Pdot`. If `N Q=(x(1+y^2))/(1+x^2)` , then the equation of curve given that it passes through the point `(3,1)` is

Options

A. `x^(2)-y^(2)=8`
B. `x^(2)+2y^(2)=11`
C. `x^(2)-5y^(2)=4`
D. None of these

Explanation

To solve the problem, we need to find the equation of the curve given the relationship between the normal at point \( P(x, y) \) and the x-axis, as well as the condition that the curve passes through the point \( (3, 1) \).

### Step-by-Step Solution:

1. **Equation of the Normal:**
The equation of the normal at point \( P(x, y) \) on the curve can be expressed as:
\[ y - y_1 = -\frac{dy}{dx}(x - x_1) \]
where \( (x_1, y_1) = (x, y) \). Thus, we have:
\[ y - y = -\frac{dy}{dx}(x - x) \]
This simplifies to:
\[ y - y = -\frac{dy}{dx}(x - x) \quad \text{(which is trivially true)} \]

2. **Finding the Length \( NQ \):**
Given that \( NQ = \frac{x(1 + y^2)}{1 + x^2} \), we can also express \( NQ \) in terms of \( y \):
\[ NQ = y \frac{dy}{dx} \]

3. **Setting the Equations Equal:**
From the above, we have:
\[ y \frac{dy}{dx} = \frac{x(1 + y^2)}{1 + x^2} \]

4. **Rearranging the Equation:**
We can rearrange this to isolate the derivatives:
\[ \frac{dy}{dx} = \frac{x(1 + y^2)}{y(1 + x^2)} \]

5. **Separating Variables:**
We can separate the variables for integration:
\[ y(1 + x^2) dy = x(1 + y^2) dx \]

6. **Integrating Both Sides:**
Integrate both sides:
\[ \int y(1 + x^2) dy = \int x(1 + y^2) dx \]
This gives:
\[ \frac{y^2}{2} + \frac{y^2 x^2}{2} = \frac{x^2}{2} + \frac{x^2 y^2}{2} + C \]

7. **Finding the Constant \( C \):**
We know the curve passes through the point \( (3, 1) \). Substitute \( x = 3 \) and \( y = 1 \) into the integrated equation to find \( C \):
\[ \frac{1^2}{2} + \frac{1^2 \cdot 3^2}{2} = \frac{3^2}{2} + \frac{3^2 \cdot 1^2}{2} + C \]
Simplifying gives:
\[ \frac{1}{2} + \frac{9}{2} = \frac{9}{2} + \frac{9}{2} + C \]
Thus:
\[ 5 = 9 + C \implies C = -4 \]

8. **Final Equation of the Curve:**
Substitute \( C \) back into the equation:
\[ \frac{y^2}{2} + \frac{y^2 x^2}{2} = \frac{x^2}{2} + \frac{x^2 y^2}{2} - 4 \]
Rearranging gives:
\[ x^2 - 5y^2 = 4 \]

### Final Answer:
The equation of the curve is:
\[ x^2 - 5y^2 = 4 \]