The radius of the director circle of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` is

Options

A. `(a-b)`
B. `sqrt(a-b)`
C. `sqrt(a^(2)-b^(2))`
D. `sqrt(a^(2)+b^(2))`

Answer

To find the radius of the director circle of the hyperbola given by the equation

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \]

we can follow these steps:

### Step 1: Understand the Director Circle
The director circle of a hyperbola is defined as the locus of the points where two perpendicular tangents to the hyperbola intersect.

### Step 2: Equation of the Director Circle
For the hyperbola given, the equation of the director circle can be derived. The general form of the equation for the director circle of a hyperbola is:

\[ x^2 + y^2 = a^2 - b^2. \]

### Step 3: Compare with the Circle Equation
The standard equation of a circle is:

\[ x^2 + y^2 = r^2, \]

where \( r \) is the radius of the circle. By comparing the two equations, we can identify that:

\[ r^2 = a^2 - b^2. \]

### Step 4: Solve for the Radius
To find the radius \( r \), we take the square root of both sides:

\[ r = \sqrt{a^2 - b^2}. \]

### Conclusion
Thus, the radius of the director circle of the hyperbola is:

\[ \sqrt{a^2 - b^2}. \]