Angle between tangents drawn from any point on the circle `x^2 +y^2 = (a + b)^2` , to the ellipse `x^2/a+y^2/b=(a+b)` is-

Options

A. `(pi)/(4)`
B. `"tan"^(-1)(1)/(2)`
C. `"tan"^(-1)(1)/(3)`
D. none of these

Answer

To find the angle between the tangents drawn from any point on the circle \( x^2 + y^2 = (a + b)^2 \) to the ellipse \( \frac{x^2}{a} + \frac{y^2}{b} = a + b \), we can follow these steps:

### Step 1: Understand the given equations
The first equation represents a circle centered at the origin with a radius of \( a + b \). The second equation represents an ellipse.

### Step 2: Rewrite the ellipse in standard form
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
To convert the given ellipse equation \( \frac{x^2}{a} + \frac{y^2}{b} = a + b \) into standard form, we can divide through by \( a + b \):
\[ \frac{x^2}{a(a+b)} + \frac{y^2}{b(a+b)} = 1 \]
Here, we can identify \( a^2 = a(a+b) \) and \( b^2 = b(a+b) \).

### Step 3: Find 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 \( a^2 \) and \( b^2 \) we found:
\[ a^2 + b^2 = a(a+b) + b(a+b) = (a + b)(a + b) = (a + b)^2 \]
Thus, the director circle of the ellipse is:
\[ x^2 + y^2 = (a + b)^2 \]

### Step 4: Analyze the tangents from the circle to the ellipse
Since the circle \( x^2 + y^2 = (a + b)^2 \) is the same as the director circle of the ellipse, any point on this circle will have tangents to the ellipse that are perpendicular to each other.

### Step 5: Conclusion about the angle between the tangents
The angle between the tangents drawn from any point on the director circle to the ellipse is \( 90^\circ \). Therefore, the angle between the tangents is:
\[ \theta = 90^\circ \]

### Final Answer
The angle between the tangents drawn from any point on the circle to the ellipse is \( 90^\circ \).

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