Number of point (s) outside the hyperbola `x^(2)/25 - y^(2)/36 = 1` from where two perpendicular tangents can be drawn to the hyperbola is (are)

Options

A. 3
B. 2
C. 1
D. 0

Answer

To find the number of points outside the hyperbola \( \frac{x^2}{25} - \frac{y^2}{36} = 1 \) from where two perpendicular tangents can be drawn, we can follow these steps:

### Step 1: Identify the parameters of the hyperbola
The given hyperbola is in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where:
- \( a^2 = 25 \) (thus \( a = 5 \))
- \( b^2 = 36 \) (thus \( b = 6 \))

### Step 2: Determine the condition for perpendicular tangents
For a hyperbola, the condition for two perpendicular tangents to exist from a point outside the hyperbola is that the point must lie on the director circle of the hyperbola. The equation of the director circle is given by:
\[ x^2 + y^2 = a^2 - b^2 \]

### Step 3: Calculate \( a^2 - b^2 \)
Now, we calculate \( a^2 - b^2 \):
\[ a^2 - b^2 = 25 - 36 = -11 \]

### Step 4: Analyze the equation of the director circle
The equation \( x^2 + y^2 = -11 \) does not represent any real points because the left side (which is a sum of squares) cannot be negative. Therefore, there are no points that satisfy this equation.

### Step 5: Conclusion
Since there are no points from which two perpendicular tangents can be drawn to the hyperbola, the answer is:
\[ \text{Number of points} = 0 \]

### Final Answer
The number of points outside the hyperbola from where two perpendicular tangents can be drawn is **0**.

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