If the sum of the slopes of the normal from a point `P` to the hyperbola `x y=c^2` is equal to `lambda(lambda in R^+)` , then the locus of point `P` is (a) `x^2=lambdac^2` (b) `y^2=lambdac^2` (c) `x y=lambdac^2` (d) none of these

Explanation

To solve the problem, we need to find the locus of the point \( P(h, k) \) such that the sum of the slopes of the normals from this point to the hyperbola \( xy = c^2 \) equals \( \lambda \), where \( \lambda \) is a positive real number.

### Step-by-step Solution:

1. **Equation of the Hyperbola**: The hyperbola we are dealing with is given by \( xy = c^2 \).

2. **Point on the Hyperbola**: Let \( (x_0, y_0) \) be a point on the hyperbola. We can parameterize this point using \( t \) as follows:
\[ x_0 = ct, \quad y_0 = \frac{c^2}{x_0} = \frac{c}{t} \]

3. **Equation of the Normal**: The equation of the normal to the hyperbola at the point \( (ct, \frac{c^2}{t}) \) is given by:
\[ ct^4 - xt^3 + ty - c = 0 \]
This is derived from the general form of the normal to the hyperbola.

4. **Slope of the Normal**: The slope of the normal at this point is given by:
\[ m = t^2 \]

5. **Normal from Point \( P(h, k) \)**: The normal line passing through the point \( P(h, k) \) can be rewritten by substituting \( x \) and \( y \) with \( h \) and \( k \):
\[ ct^4 - ht^3 + tk - c = 0 \]

6. **Roots of the Normal**: Let the slopes of the normals from point \( P(h, k) \) be \( t_1, t_2 \). The sum of the slopes of the normals is:
\[ t_1 + t_2 = \frac{h}{c} \]
The product of the slopes is:
\[ t_1 t_2 = 0 \]

7. **Sum of the Slopes**: The sum of the squares of the slopes is given by:
\[ t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1 t_2 = \left(\frac{h}{c}\right)^2 - 0 = \frac{h^2}{c^2} \]

8. **Setting Equal to Lambda**: According to the problem, the sum of the slopes of the normals is equal to \( \lambda \):
\[ t_1^2 + t_2^2 = \lambda \]
Therefore, we have:
\[ \frac{h^2}{c^2} = \lambda \]

9. **Finding the Locus**: Rearranging the above equation gives:
\[ h^2 = \lambda c^2 \]
In terms of coordinates, this can be expressed as:
\[ x^2 = \lambda c^2 \]

### Conclusion:
The locus of the point \( P \) is given by:
\[ x^2 = \lambda c^2 \]

Thus, the correct option is (a) \( x^2 = \lambda c^2 \).