The product of the perpendicular from two foci on any tangent to the hyperbola `x^2/a^2-y^2/b^2=1` is (A) `a^2` (B) `(b/a)^2` (C) `(a/b)^2` (D) `b^2`

Options

A. `b^(2)`
B. `2b^(2)`
C. `a^(2)`
D. `2a^(2)`

Explanation

To solve the problem, we need to find the product of the perpendicular distances from the two foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) to any tangent line to the hyperbola. Let's go through the steps systematically.

### Step 1: Identify the foci of the hyperbola
The foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are located at the points \( (ae, 0) \) and \( (-ae, 0) \), where \( e = \sqrt{1 + \frac{b^2}{a^2}} \).

**Hint:** Remember that the foci of a hyperbola are determined using the formula \( e = \sqrt{1 + \frac{b^2}{a^2}} \).

### Step 2: Write the equation of the tangent
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