Tangents drawn from a point on the circle `x^2+y^2=9` to the hyperbola `x^2/25-y^2/16=1,`
then tangents are at angle

Options

A. (a) `(pi)/(6)`
B. (b) `(pi)/(4)`
C. (c) `(pi)/(3)`
D. (d) `(pi)/(2)`

Answer

To solve the problem of finding the angle between the tangents drawn from a point on the circle \( x^2 + y^2 = 9 \) to the hyperbola \( \frac{x^2}{25} - \frac{y^2}{16} = 1 \), we can follow these steps:

### Step 1: Identify the equations
We have the equation of the circle:
\[ x^2 + y^2 = 9 \]
and the equation of the hyperbola:
\[ \frac{x^2}{25} - \frac{y^2}{16} = 1 \]

### Step 2: Identify parameters of the hyperbola
From the hyperbola's equation, we can identify:
- \( a^2 = 25 \) which gives \( a = 5 \)
- \( b^2 = 16 \) which gives \( b = 4 \)

### Step 3: Find the equation of the director circle
The equation of the director circle for the hyperbola is given by:
\[ x^2 + y^2 = a^2 - b^2 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ x^2 + y^2 = 25 - 16 = 9 \]

### Step 4: Compare with the given circle
The equation of the director circle is:
\[ x^2 + y^2 = 9 \]
This is the same as the equation of the given circle \( x^2 + y^2 = 9 \).

### Step 5: Conclusion about tangents
Since the given circle is the director circle of the hyperbola, it implies that any point on this circle will have tangents to the hyperbola that are perpendicular to each other.

### Step 6: Determine the angle between the tangents
When tangents are perpendicular, the angle between them is:
\[ \theta = \frac{\pi}{2} \]

### Final Answer
Thus, the angle between the tangents drawn from a point on the circle to the hyperbola is:
\[ \text{Option D: } \frac{\pi}{2} \]

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