The product of perpendicular drawn from any points on a hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` 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)/(ab)`

Answer

To find the product of the perpendicular distances drawn from any point on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) to its asymptotes, we can follow these steps:

### Step 1: Identify the asymptotes of the hyperbola
The asymptotes of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are given by the equations:
\[ y = \frac{b}{a}x \quad \text{and} \quad y = -\frac{b}{a}x \]

### Step 2: Choose a point on the hyperbola
Let \(P(x_1, y_1)\) be a point on the hyperbola. This point satisfies the hyperbola equation:
\[ \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = 1 \]

### Step 3: Calculate the distances from the point to the asymptotes
The distances from the point \(P(x_1, y_1)\) to the asymptotes can be calculated using the formula for the distance from a point to a line.

For the first asymptote \(y = \frac{b}{a}x\), we can rewrite it in standard form:
\[ \frac{b}{a}x - y = 0 \quad \Rightarrow \quad bx - ay = 0 \]
The distance \(d_1\) from point \(P(x_1, y_1)\) to this line is:
\[ d_1 = \frac{|bx_1 - ay_1|}{\sqrt{b^2 + a^2}} \]

For the second asymptote \(y = -\frac{b}{a}x\), we rewrite it as:
\[ -\frac{b}{a}x - y = 0 \quad \Rightarrow \quad -bx - ay = 0 \]
The distance \(d_2\) from point \(P(x_1, y_1)\) to this line is:
\[ d_2 = \frac{|-bx_1 - ay_1|}{\sqrt{b^2 + a^2}} \]

### Step 4: Calculate the product of the distances
Now, we need to find the product \(d_1 \cdot d_2\):
\[ d_1 \cdot d_2 = \left(\frac{|bx_1 - ay_1|}{\sqrt{b^2 + a^2}}\right) \cdot \left(\frac{|-bx_1 - ay_1|}{\sqrt{b^2 + a^2}}\right) \]
\[ = \frac{|bx_1 - ay_1| \cdot |-bx_1 - ay_1|}{b^2 + a^2} \]

### Step 5: Simplify the expression
Using the property of absolute values, we can simplify:
\[ |bx_1 - ay_1| \cdot |-bx_1 - ay_1| = |(bx_1 - ay_1)(-bx_1 - ay_1)| = |-(bx_1)^2 + (ay_1)^2| = |a^2y_1^2 - b^2x_1^2| \]
Since \(x_1\) and \(y_1\) satisfy the hyperbola equation, we can substitute \(b^2x_1^2 - a^2y_1^2 = -a^2b^2\):
\[ = \frac{|-a^2b^2|}{b^2 + a^2} \]

### Final Result
Thus, the product of the perpendicular distances from the point \(P(x_1, y_1)\) to the asymptotes is:
\[ \frac{a^2b^2}{a^2 + b^2} \]