If the common tangets of `x^(2)+y^(2)=r^(2) and (x^(2))/(16)+(y^(2))/(9)=1` form a square, then the area (in sq. units) of the square is

Options

A. 50
B. 100
C. 25
D. 40

Answer

To solve the problem, we need to find the area of the square formed by the common tangents of the given circle and ellipse. Let's break this down step by step.

### Step 1: Identify the equations of the shapes
The equations given are:
1. Circle: \( x^2 + y^2 = r^2 \)
2. Ellipse: \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \)

### Step 2: Find the semi-major and semi-minor axes of the ellipse
From the ellipse equation, we can identify:
- Semi-major axis \( a = 4 \) (since \( 16 = 4^2 \))
- Semi-minor axis \( b = 3 \) (since \( 9 = 3^2 \))

### Step 3: Find the equation of the director circle
The director circle of a circle with radius \( r \) is given by:
\[ x^2 + y^2 = 2r^2 \]

The director circle of the ellipse is given by:
\[ x^2 + y^2 = a^2 + b^2 \]
Substituting the values of \( a \) and \( b \):
\[ a^2 + b^2 = 16 + 9 = 25 \]

### Step 4: Set the equations equal to find \( r \)
Since the director circles of both the circle and the ellipse are equal, we have:
\[ 2r^2 = 25 \]
From this, we can solve for \( r^2 \):
\[ r^2 = \frac{25}{2} \]

### Step 5: Find the radius of the director circle
Taking the square root gives us:
\[ r = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \]

### Step 6: Find the side length of the square
The side length \( s \) of the square formed by the tangents is equal to the radius of the director circle:
\[ s = r\sqrt{2} = \frac{5\sqrt{2}}{2} \cdot \sqrt{2} = \frac{5 \cdot 2}{2} = 5 \]

### Step 7: Calculate the area of the square
The area \( A \) of the square is given by:
\[ A = s^2 = 5^2 = 25 \]

### Final Answer
Thus, the area of the square formed by the common tangents is:
\[ \boxed{25} \text{ square units} \]