the product of the perpendicular distance from any points on a hyperbola to its asymptotes is

Options

A. `(ab)/((sqrt(a)+sqrt(b))`
B. `(ab)/(a^2+b^2)`
C. `(a^2b^2)/(a^2+b^2)`
D. `(a^2+b^2)/(a^2b^2)`

Answer

To solve the problem of finding the product of the perpendicular distances from any point on a hyperbola to its asymptotes, we can follow these steps:

### Step 1: Define the Hyperbola
We start with the standard form of the hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This represents a hyperbola centered at the origin with transverse axis along the x-axis.

### Step 2: Determine the Asymptotes
The asymptotes of the hyperbola can be derived from the equation by considering the highest degree terms:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \]
This can be factored into two linear equations:
\[ bx - ay = 0 \quad \text{and} \quad bx + ay = 0 \]
Thus, the equations of the asymptotes are:
1. \( y = \frac{b}{a}x \)
2. \( y = -\frac{b}{a}x \)

### Step 3: Choose a Point on the Hyperbola
Let’s consider a general point on the hyperbola given by:
\[ P(a \sec \theta, b \tan \theta) \]
where \( \theta \) is a parameter.

### Step 4: Calculate the Perpendicular Distances
We will use the formula for the perpendicular distance from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \):
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

#### Distance from the first asymptote \( bx - ay = 0 \):
- Here, \( A = b \), \( B = -a \), and \( C = 0 \).
- For point \( P(a \sec \theta, b \tan \theta) \):
\[ d_1 = \frac{|b(a \sec \theta) - a(b \tan \theta)|}{\sqrt{b^2 + a^2}} = \frac{|ab \sec \theta - ab \tan \theta|}{\sqrt{b^2 + a^2}} = \frac{ab(\sec \theta - \tan \theta)}{\sqrt{a^2 + b^2}} \]

#### Distance from the second asymptote \( bx + ay = 0 \):
- Here, \( A = b \), \( B = a \), and \( C = 0 \).
- For point \( P(a \sec \theta, b \tan \theta) \):
\[ d_2 = \frac{|b(a \sec \theta) + a(b \tan \theta)|}{\sqrt{b^2 + a^2}} = \frac{|ab \sec \theta + ab \tan \theta|}{\sqrt{b^2 + a^2}} = \frac{ab(\sec \theta + \tan \theta)}{\sqrt{a^2 + b^2}} \]

### Step 5: Calculate the Product of Distances
Now, we multiply \( d_1 \) and \( d_2 \):
\[ d_1 \cdot d_2 = \left(\frac{ab(\sec \theta - \tan \theta)}{\sqrt{a^2 + b^2}}\right) \cdot \left(\frac{ab(\sec \theta + \tan \theta)}{\sqrt{a^2 + b^2}}\right) \]
This simplifies to:
\[ = \frac{(ab)^2(\sec^2 \theta - \tan^2 \theta)}{a^2 + b^2} \]
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ = \frac{(ab)^2}{a^2 + b^2} \]

### Final Result
Thus, the product of the perpendicular distances from any point on the hyperbola to its asymptotes is:
\[ \frac{a^2 b^2}{a^2 + b^2} \]