IF the locus of the point of intersection of two perpendicular tangents to a hyperbola `(x^(2))/(25) - (y^(2))/(16) =1` is a circle with centre (0, 0), then the radius of a circle is

Options

A. 5
B. 4
C. 3
D. 7

Answer

To solve the problem, we need to find the radius of the circle that represents the locus of the point of intersection of two perpendicular tangents to the given hyperbola. The hyperbola in question is given by the equation:

\[ \frac{x^2}{25} - \frac{y^2}{16} = 1 \]

### Step 1: Identify \( a^2 \) and \( b^2 \)
From the hyperbola's equation, we can identify:
- \( a^2 = 25 \)
- \( b^2 = 16 \)

### Step 2: Use the formula for the director circle
The locus of the point of intersection of two perpendicular tangents to a hyperbola is known as the director circle. The equation of the director circle is given by:

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

### Step 3: Substitute the values of \( a^2 \) and \( b^2 \)
Now, substituting the values we found into the equation:

\[ x^2 + y^2 = 25 - 16 \]

### Step 4: Simplify the equation
Calculating the right-hand side:

\[ x^2 + y^2 = 9 \]

### Step 5: Identify the radius of the circle
The equation \( x^2 + y^2 = 9 \) represents a circle centered at the origin (0, 0) with a radius \( r \) given by:

\[ r^2 = 9 \implies r = \sqrt{9} = 3 \]

Thus, the radius of the circle is:

\[ \text{Radius} = 3 \]

### Conclusion
The radius of the circle is \( 3 \).

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