Find the points on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))= 2` from which two perpendicular tangents can be drawn to the circle `x^(2) + y^(2) = a^(2)`

Answer

To solve the problem of finding the points on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 2\) from which two perpendicular tangents can be drawn to the circle \(x^2 + y^2 = a^2\), we will follow these steps:

### Step 1: Rewrite the Hyperbola in Standard Form
We start with the hyperbola equation:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 2 \]
To rewrite it in standard form, we can multiply through by \( \frac{1}{2} \):
\[ \frac{x^2}{2a^2} - \frac{y^2}{2b^2} = 1 \]
This shows that the hyperbola has semi-major axis \( \sqrt{2}a \) and semi-minor axis \( \sqrt{2}b \).

### Step 2: Equation of Tangents to the Circle
The equation of the circle is given by:
\[ x^2 + y^2 = a^2 \]
The equation of the tangent to the circle at point \((h, k)\) can be expressed as:
\[ y - k = m(x - h) \]
where \(m\) is the slope of the tangent.

### Step 3: Condition for Perpendicular Tangents
For two tangents to be perpendicular, the product of their slopes \(m_1\) and \(m_2\) must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]

### Step 4: Substitute the Point into the Circle Equation
The point \((h, k)\) must satisfy the circle's equation:
\[ h^2 + k^2 = a^2 \]

### Step 5: Substitute into the Tangent Equation
Substituting the tangent equation into the circle's equation leads to:
\[ y^2 + m^2x^2 - 2mxy = a^2(1 + m^2) \]
Rearranging gives us a quadratic in \(x\):
\[ (1 + m^2)x^2 - 2mxy + (y^2 - a^2) = 0 \]

### Step 6: Find the Condition for Perpendicular Tangents
The condition for the tangents to be perpendicular is derived from the discriminant of the quadratic equation:
\[ D = (-2my)^2 - 4(1 + m^2)(y^2 - a^2) = 0 \]
This leads to:
\[ 4m^2y^2 - 4(1 + m^2)(y^2 - a^2) = 0 \]
Simplifying gives us:
\[ m^2y^2 = (1 + m^2)(y^2 - a^2) \]

### Step 7: Solve for \(y\)
From the above equation, we can isolate \(y\) and find the relationship between \(y\) and \(a\):
\[ m^2y^2 = y^2 + m^2a^2 - y^2 \implies (m^2 - 1)y^2 = m^2a^2 \]
This implies:
\[ y^2 = \frac{m^2a^2}{m^2 - 1} \]

### Step 8: Substitute Back into the Hyperbola
Now we substitute \(y^2\) back into the hyperbola equation to find \(x\):
\[ \frac{x^2}{a^2} - \frac{\frac{m^2a^2}{m^2 - 1}}{b^2} = 2 \]
This will give us the values of \(x\) in terms of \(m\).

### Step 9: Analyze the Result
We analyze the results to determine how many points satisfy both the hyperbola and the condition for perpendicular tangents.

### Conclusion
After analyzing the conditions, we find that there are no points on the hyperbola from which two perpendicular tangents can be drawn to the circle.