The product of lengths of perpendicular from any point on the hyperola `x^(2)-y^(2)=16` to its asymptotes is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To find the product of the lengths of the perpendiculars from any point on the hyperbola \( x^2 - y^2 = 16 \) to its asymptotes, we will follow these steps:

### Step 1: Rewrite the Hyperbola in Standard Form
The given hyperbola is \( x^2 - y^2 = 16 \). We can rewrite it in standard form by dividing both sides by 16:

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

This gives us \( a^2 = 16 \) and \( b^2 = 16 \), so \( a = 4 \) and \( b = 4 \).

### Step 2: Determine the Asymptotes
The asymptotes of a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by the equations:

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

Substituting \( a = 4 \) and \( b = 4 \):

\[ y = \pm 1 x \quad \text{or} \quad y = \pm x \]

Thus, the equations of the asymptotes are \( y = x \) and \( y = -x \).

### Step 3: General Point on the Hyperbola
A general point on the hyperbola can be expressed as:

\[ P(a \sec \theta, b \tan \theta) = (4 \sec \theta, 4 \tan \theta) \]

### Step 4: Calculate the Perpendicular Distances
1. **Distance from \( y = x \)**:
The equation of the line \( y = x \) can be rewritten as \( x - y = 0 \). The perpendicular distance \( p_1 \) from the point \( P(4 \sec \theta, 4 \tan \theta) \) to this line is given by:

\[ p_1 = \frac{|4 \sec \theta - 4 \tan \theta|}{\sqrt{1^2 + (-1)^2}} = \frac{4 |\sec \theta - \tan \theta|}{\sqrt{2}} = \frac{4}{\sqrt{2}} |\sec \theta - \tan \theta| \]

2. **Distance from \( y = -x \)**:
The equation of the line \( y = -x \) can be rewritten as \( x + y = 0 \). The perpendicular distance \( p_2 \) from the point \( P(4 \sec \theta, 4 \tan \theta) \) to this line is given by:

\[ p_2 = \frac{|4 \sec \theta + 4 \tan \theta|}{\sqrt{1^2 + 1^2}} = \frac{4 |\sec \theta + \tan \theta|}{\sqrt{2}} = \frac{4}{\sqrt{2}} |\sec \theta + \tan \theta| \]

### Step 5: Calculate the Product of the Distances
Now we need to find the product \( p_1 \cdot p_2 \):

\[ p_1 \cdot p_2 = \left( \frac{4}{\sqrt{2}} |\sec \theta - \tan \theta| \right) \cdot \left( \frac{4}{\sqrt{2}} |\sec \theta + \tan \theta| \right) \]

This simplifies to:

\[ = \frac{16}{2} |\sec \theta - \tan \theta| |\sec \theta + \tan \theta| = 8 |\sec^2 \theta - \tan^2 \theta| \]

Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):

\[ p_1 \cdot p_2 = 8 \cdot 1 = 8 \]

### Final Answer
Thus, the product of the lengths of the perpendiculars from any point on the hyperbola to its asymptotes is:

\[ \boxed{8} \]