For hyperbola `x^2/a^2-y^2/b^2=1` , let n be the number of points on the plane through which perpendicular tangents are drawn.

Options

A. If n = 1, then `e=sqrt2`
B. If n gt 1, then `0 lt e lt sqrt2.`
C. If n = 0, then `e gt sqrt2`.
D. none of these

Explanation

To solve the problem of finding the number of points on the plane through which perpendicular tangents can be drawn to the hyperbola given by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we will analyze the conditions under which these tangents can be perpendicular.

### Step-by-Step Solution:

1. **Understanding the Hyperbola:**
The given hyperbola is defined by the equation:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This is a standard form of a hyperbola that opens horizontally.

2. **Equation of Tangents:**
The equation of the tangent to the hyperbola at a point \((x_0, y_0)\) on the hyperbola can be expressed as:
\[ \frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1 \]

3. **Condition for Perpendicular Tangents:**
For two tangents to be perpendicular, the product of their slopes must equal -1. If we denote the slopes of the tangents as \(m_1\) and \(m_2\), the condition for perpendicularity is:
\[ m_1 \cdot m_2 = -1 \]

4. **Finding the Slopes:**
The slopes of the tangents from a point \((x_1, y_1)\) to the hyperbola can be derived from the tangent equation. The slopes can be found using the quadratic equation formed by substituting \(y = mx + c\) into the hyperbola equation.

5. **Using the Discriminant:**
The tangents will be real if the discriminant of the resulting quadratic equation is non-negative. For perpendicular tangents, we need to analyze the conditions under which the slopes satisfy the perpendicularity condition.

6. **Analyzing Cases:**
We will consider three cases based on the relationship between \(a\) and \(b\):
- **Case 1:** \(a^2 - b^2 > 0\) (real tangents exist)
- **Case 2:** \(a^2 - b^2 = 0\) (one point of tangency)
- **Case 3:** \(a^2 - b^2 < 0\) (no real tangents)

7. **Conclusion on Points:**
- For **Case 1**, there are infinitely many points from which perpendicular tangents can be drawn (i.e., \(n = \infty\)).
- For **Case 2**, there is exactly one point from which the tangents are perpendicular (i.e., \(n = 1\)).
- For **Case 3**, there are no points from which tangents can be drawn (i.e., \(n = 0\)).

### Final Result:
Thus, the number of points \(n\) on the plane through which perpendicular tangents can be drawn to the hyperbola can take values of \(0\), \(1\), or \(\infty\) depending on the relationship between \(a\) and \(b\).