The number of points on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=3` from which mutually perpendicular tangents can be drawn to the circle `x^2+y^2=a^2` is/are (a) 0 (b) 2 (c) 3 (d) 4

Options

A. 0
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to determine the number of points on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 3\) from which mutually perpendicular tangents can be drawn to the circle \(x^2 + y^2 = a^2\).

### Step-by-Step Solution:

1. **Identify the Circle and its Director Circle**:
The equation of the circle is given by \(x^2 + y^2 = a^2\). The radius of this circle is \(a\).
The director circle for this situation, from which tangents can be drawn, is given by the equation \(x^2 + y^2 = 2a^2\).

2. **Rewrite the Hyperbola Equation**:
The hyperbola is given as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 3\). We can rewrite this in a more standard form:
\[ \frac{x^2}{3a^2} - \frac{y^2}{3b^2} = 1 \]
From this, we can identify the semi-transverse axis \(A = \sqrt{3}a\) and semi-conjugate axis \(B = \sqrt{3}b\).

3. **Locate the Points of Interest**:
The points from which we can draw tangents to the circle must lie outside the circle of radius \(a\) and also satisfy the condition of being on the hyperbola. The points on the hyperbola can be represented as \((x, y)\) such that they satisfy the hyperbola equation.

4. **Condition for Mutually Perpendicular Tangents**:
For the tangents to be mutually perpendicular, the point from which the tangents are drawn must lie on the director circle. The director circle's equation is \(x^2 + y^2 = 2a^2\).

5. **Finding Intersection Points**:
We need to find the intersection points of the hyperbola and the director circle:
- The hyperbola: \(\frac{x^2}{3a^2} - \frac{y^2}{3b^2} = 1\)
- The director circle: \(x^2 + y^2 = 2a^2\)

Substitute \(y^2\) from the director circle into the hyperbola equation:
\[ y^2 = 2a^2 - x^2 \]
Substitute into the hyperbola equation:
\[ \frac{x^2}{3a^2} - \frac{2a^2 - x^2}{3b^2} = 1 \]
Simplifying this will yield a quadratic in \(x^2\).

6. **Analyzing the Quadratic**:
The discriminant of this quadratic will determine the number of real solutions (points of intersection). If the discriminant is negative, there are no intersection points.

7. **Conclusion**:
After analyzing the intersection, we find that there are no points of intersection between the hyperbola and the director circle. Therefore, the number of points on the hyperbola from which mutually perpendicular tangents can be drawn to the circle is **0**.

### Final Answer:
Thus, the answer is \(0\).