For which of the hyperbolas, can we have more than one pair of perpendicular tangents? `(x^2)/4-(y^2)/9=1` (b) `(x^2)/4-(y^2)/9=-1` `x^2-y^2=4` (d) `x y=44`

Options

A. `(x^(2))/(4)-(y^(2))/(9)=1`
B. `(x^(2))/(4)-(y^(2))/(9)=-1`
C. `x^(2)-y^(2)=4`
D. `xy=44`

Explanation

To determine which hyperbolas can have more than one pair of perpendicular tangents, we need to analyze the given equations using the properties of hyperbolas and their director circles.

### Step-by-Step Solution:

1. **Identify the General Form of Hyperbola**:
The general form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
or
\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]

2. **Understand the Condition for Perpendicular Tangents**:
For a hyperbola, the intersection point of two perpendicular tangents lies on the director circle. The equation of the director circle is given by:
\[ x^2 + y^2 = a^2 - b^2 \]
This condition holds true when \( a^2 > b^2 \).

3. **Analyze Each Option**:
- **Option (a)**: \(\frac{x^2}{4} - \frac{y^2}{9} = 1\)
- Here, \( a^2 = 4 \) and \( b^2 = 9 \). Since \( 4 < 9 \), this hyperbola does not satisfy the condition for perpendicular tangents.

- **Option (b)**: \(\frac{x^2}{4} - \frac{y^2}{9} = -1\)
- This represents a hyperbola in the form \(\frac{y^2}{9} - \frac{x^2}{4} = 1\). Here, \( b^2 = 9 \) and \( a^2 = 4 \). Since \( 9 > 4 \), this hyperbola can have perpendicular tangents.

- **Option (c)**: \(x^2 - y^2 = 4\)
- This can be rewritten as \(\frac{x^2}{4} - \frac{y^2}{4} = 1\). Here, \( a^2 = 4 \) and \( b^2 = 4 \). Since \( a^2 = b^2 \), this hyperbola does not satisfy the condition for perpendicular tangents.

- **Option (d)**: \(xy = 44\)
- This is a rectangular hyperbola, which can be expressed in the form \(\frac{x^2}{44} - \frac{y^2}{44} = 1\). Here, \( a^2 = 44 \) and \( b^2 = 44 \). Since \( a^2 = b^2 \), this hyperbola does not satisfy the condition for perpendicular tangents.

4. **Conclusion**:
The only hyperbola that can have more than one pair of perpendicular tangents is option (b):
\[ \frac{x^2}{4} - \frac{y^2}{9} = -1 \]