The number of points outside the hyperbola `x^2/9-y^2/16=1` from where two perpendicular tangents can be drawn to the hyperbola are: (a) 0 (b) 1 (c) 2 (d) none of these

Options

A. `0`
B. `1`
C. `2`
D. None of these

Answer

To solve the problem, we need to determine the number of points outside the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) from which two perpendicular tangents can be drawn. Here’s the step-by-step solution:

### Step 1: Identify the hyperbola parameters
The given hyperbola is in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where:
- \( a^2 = 9 \) → \( a = 3 \)
- \( b^2 = 16 \) → \( b = 4 \)

### Step 2: Determine the equation of the director circle
For a hyperbola, the equation of the director circle is given by:
\[ x^2 + y^2 = a^2 - b^2 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ x^2 + y^2 = 9 - 16 = -7 \]

### Step 3: Analyze the director circle equation
The equation \( x^2 + y^2 = -7 \) indicates that the left-hand side (which represents the sum of squares) cannot equal a negative number. This implies that there are no real points \( (x, y) \) that satisfy this equation.

### Step 4: Conclusion
Since the director circle does not have any real solutions, it means that there are no points outside the hyperbola from which two perpendicular tangents can be drawn.

Thus, the number of such points is:
\[ \text{Answer: } 0 \]

### Final Answer
The correct option is (a) 0.

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