Tangents are drawn from any point on the circle `x^(2)+y^(2) = 41` to the ellipse `(x^(2))/(25)+(y^(2))/(16) =1` then the angle between the two tangents is

Options

A. `(pi)/(4)`
B. `(pi)/(3)`
C. `(pi)/(6)`
D. `(pi)/(2)`

Explanation

To solve the problem, we need to find the angle between the tangents drawn from any point on the circle \( x^2 + y^2 = 41 \) to the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).

### Step 1: Identify the Circle and Ellipse Parameters
The equation of the circle is given by:
\[ x^2 + y^2 = 41 \]
This can be rewritten as:
\[ x^2 + y^2 = (\sqrt{41})^2 \]
This indicates that the radius of the circle is \( \sqrt{41} \).

The equation of the ellipse is given by:
\[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \]
From this, we can identify the semi-major axis \( a = 5 \) and the semi-minor axis \( b = 4 \).

### Step 2: Determine the Director Circle of the Ellipse
The director circle of an ellipse is given by the equation:
\[ x^2 + y^2 = a^2 + b^2 \]
Substituting the values of \( a \) and \( b \):
\[ a^2 = 25, \quad b^2 = 16 \]
Thus,
\[ a^2 + b^2 = 25 + 16 = 41 \]
Therefore, the equation of the director circle is:
\[ x^2 + y^2 = 41 \]
This is the same as the equation of the given circle.

### Step 3: Understand the Tangents from the Director Circle
It is a known property that tangents drawn from any point on the director circle of an ellipse to the ellipse are perpendicular to each other. Therefore, the angle between the two tangents is \( 90^\circ \).

### Step 4: Conclusion
Thus, the angle between the two tangents drawn from any point on the circle \( x^2 + y^2 = 41 \) to the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) is:
\[ \text{Angle} = 90^\circ = \frac{\pi}{2} \text{ radians} \]

### Final Answer
The angle between the two tangents is \( \frac{\pi}{2} \) radians.

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