If the sum of the slopes of the normals from a point P to the hyperbola `xy=c ^(2)` is constant `k (k gt 0),` then the locus of point P is

Options

A. `x ^(2) = kc ^(2)`
B. `x ^(2) = kc`
C. `x ^(2) + kx ^(2) =0`
D. none of these

Answer

To solve the problem, we need to find the locus of a point \( P(h, k) \) such that the sum of the slopes of the normals from \( P \) to the hyperbola \( xy = c^2 \) is a constant \( k \) (where \( k > 0 \)).

### Step-by-step Solution:

1. **Parametric Representation of the Hyperbola:**
The hyperbola \( xy = c^2 \) can be represented parametrically as:
\[ x = ct, \quad y = \frac{c^2}{t} \]
where \( t \) is a parameter.

2. **Equation of the Normal:**
The equation of the normal to the hyperbola at the point \( (ct, \frac{c^2}{t}) \) is given by:
\[ ct^4 - x t^3 + y t - c = 0 \]
Here, \( x \) and \( y \) will be replaced by \( h \) and \( k \) respectively, as we are considering the point \( P(h, k) \).

3. **Finding the Slope of the Normal:**
The slope of the normal line can be derived from the equation above. The slope \( m \) of the normal is given by:
\[ m = -\frac{\text{coefficient of } x}{\text{coefficient of } y} = -\frac{-t^3}{t} = t^2 \]
Therefore, the slope of the normal at the point is \( t^2 \).

4. **Sum of the Slopes:**
If \( t_1, t_2, t_3, t_4 \) are the roots corresponding to the normals from point \( P(h, k) \), the sum of the slopes of the normals is:
\[ S = t_1^2 + t_2^2 + t_3^2 + t_4^2 \]

5. **Using the Sum of Roots Formula:**
By Vieta's formulas, we know:
- The sum of the roots \( t_1 + t_2 + t_3 + t_4 = \frac{h}{c} \)
- The sum of the product of the roots taken two at a time \( t_1 t_2 + t_1 t_3 + t_1 t_4 + t_2 t_3 + t_2 t_4 + t_3 t_4 = 0 \) (as there is no \( t^2 \) term in the polynomial).

6. **Calculating the Sum of Squares:**
Using the identity:
\[ t_1^2 + t_2^2 + t_3^2 + t_4^2 = (t_1 + t_2 + t_3 + t_4)^2 - 2(t_1 t_2 + t_1 t_3 + t_1 t_4 + t_2 t_3 + t_2 t_4 + t_3 t_4) \]
We substitute:
\[ t_1^2 + t_2^2 + t_3^2 + t_4^2 = \left(\frac{h}{c}\right)^2 - 2(0) = \left(\frac{h}{c}\right)^2 \]

7. **Setting the Condition:**
According to the problem, the sum of the slopes of the normals is constant \( k \):
\[ t_1^2 + t_2^2 + t_3^2 + t_4^2 = k \]
Therefore, we have:
\[ \left(\frac{h}{c}\right)^2 = k \]

8. **Finding the Locus:**
Rearranging gives us:
\[ h^2 = k c^2 \]
This represents a parabola in the \( hk \)-plane.

### Conclusion:
The locus of the point \( P(h, k) \) is given by the equation:
\[ h^2 = k c^2 \]