The product of the perpendiculars from the foci on any tangent to the hyperbol `(x^(2))/(64)-(y^(2))/(9)=1` is

Options

A. 8
B. 9
C. 16
D. 18

Answer

To solve the problem of finding the product of the perpendiculars from the foci on any tangent to the hyperbola given by the equation \(\frac{x^2}{64} - \frac{y^2}{9} = 1\), we will follow these steps:

### Step 1: Identify the parameters of the hyperbola
The standard form of the hyperbola is given as:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
From the given equation, we can identify:
- \(a^2 = 64\) → \(a = \sqrt{64} = 8\)
- \(b^2 = 9\) → \(b = \sqrt{9} = 3\)

### Step 2: Find the foci of the hyperbola
The foci of a hyperbola are located at \((\pm c, 0)\), where \(c\) is calculated using the formula:
\[ c = \sqrt{a^2 + b^2} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ c = \sqrt{64 + 9} = \sqrt{73} \]
Thus, the foci are at the points \((\sqrt{73}, 0)\) and \((- \sqrt{73}, 0)\).

### Step 3: Use the property of the hyperbola
A property of hyperbolas states that the product of the perpendiculars from the foci to any tangent line is equal to \(b^2\). Since we have already determined \(b^2\):
\[ b^2 = 9 \]

### Step 4: Conclusion
Therefore, the product of the perpendiculars from the foci on any tangent to the hyperbola is:
\[ \text{Product of perpendiculars} = 9 \]

### Final Answer:
The product of the perpendiculars from the foci on any tangent to the hyperbola \(\frac{x^2}{64} - \frac{y^2}{9} = 1\) is **9**.

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