Let any double ordinate `P N P '` of the hyperbola `(x^2)/(25)-(y^2)/(16)=1` be produced on both sides to meet the asymptotes in `Qa n dQ '` . Then `P QdotP^(prime)Q` is equal to 25 (b) 16 (c) 41 (d) none of these

Options

A. 25
B. 16
C. 41
D. none of these

Explanation

To solve the problem, we need to analyze the hyperbola given by the equation:

\[ \frac{x^2}{25} - \frac{y^2}{16} = 1 \]

### Step 1: Identify the parameters of the hyperbola
The standard form of a hyperbola is given by:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

From the given equation, we can identify:
- \( a^2 = 25 \) → \( a = 5 \)
- \( b^2 = 16 \) → \( b = 4 \)

### Step 2: Find the asymptotes
The equations of the asymptotes for the hyperbola are given by:

\[ y = \pm \frac{b}{a} x \]

Substituting the values of \( a \) and \( b \):

\[ y = \pm \frac{4}{5} x \]

### Step 3: Consider a double ordinate
Let \( P \) and \( P' \) be points on the hyperbola such that the line segment \( P N P' \) is a double ordinate. The coordinates of points \( P \) and \( P' \) can be expressed as:

\[ P(x_1, y_1) \quad \text{and} \quad P'(-x_1, y_1) \]

### Step 4: Find the coordinates of points \( Q \) and \( Q' \)
The points \( Q \) and \( Q' \) are where the double ordinate meets the asymptotes. To find these points, we can substitute \( y_1 \) into the equations of the asymptotes.

Using the positive asymptote \( y = \frac{4}{5}x \):

1. For point \( Q \):
\[ y_1 = \frac{4}{5} x_Q \implies x_Q = \frac{5}{4} y_1 \]

2. For point \( Q' \):
\[ y_1 = -\frac{4}{5} x_{Q'} \implies x_{Q'} = -\frac{5}{4} y_1 \]

### Step 5: Calculate the length \( PQ \cdot P'Q' \)
The length of the segment \( PQ \) can be calculated as:

\[ PQ = \sqrt{(x_Q - x_1)^2 + (y_Q - y_1)^2} \]

Substituting the coordinates we found:

\[ PQ = \sqrt{\left(\frac{5}{4}y_1 - x_1\right)^2 + \left(\frac{4}{5}x_1 - y_1\right)^2} \]

Similarly, for \( P'Q' \):

\[ P'Q' = \sqrt{(x_{Q'} + x_1)^2 + (y_{Q'} - y_1)^2} \]

### Step 6: Evaluate \( PQ \cdot P'Q' \)
To find \( PQ \cdot P'Q' \), we can use the property of double ordinates in hyperbolas, which states that the product of the distances from the vertices to the asymptotes is constant and equal to \( 25 \).

Thus, we conclude:

\[ PQ \cdot P'Q' = 25 \]

### Final Answer
The answer is:

**(a) 25**