If the eccentricites of the ellipse `x^(2)/4+y^(2)/3=1` and the hyperbola `x^(2)/64-y^(2)/b^(2)=1` are reciprocals of each other, then `b^(2)` is equal to

Options

A. 192
B. 64
C. 16
D. 32

Explanation

To solve the problem, we need to find the value of \( b^2 \) given that the eccentricities of the ellipse and hyperbola are reciprocals of each other.

### Step 1: Identify the parameters of the ellipse and hyperbola

The equation of the ellipse is given as:

\[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \]

From this, we can identify:

- \( a^2 = 4 \) (where \( a \) is the semi-major axis)
- \( b^2 = 3 \) (where \( b \) is the semi-minor axis)

The equation of the hyperbola is given as:

\[ \frac{x^2}{64} - \frac{y^2}{b^2} = 1 \]

From this, we can identify:

- \( a^2 = 64 \) (where \( a \) is the semi-major axis)
- \( b^2 = b^2 \) (where \( b \) is the semi-minor axis)

### Step 2: Calculate the eccentricity of the ellipse

The eccentricity \( e \) of an ellipse is given by the formula:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

Substituting the values for the ellipse:

\[ e_{ellipse} = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 3: Relate the eccentricities

According to the problem, the eccentricity of the hyperbola is the reciprocal of the eccentricity of the ellipse:

\[ e_{hyperbola} = \frac{1}{e_{ellipse}} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 4: Calculate the eccentricity of the hyperbola

The eccentricity \( e \) of a hyperbola is given by the formula:

\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]

Substituting the values for the hyperbola:

\[ e_{hyperbola} = \sqrt{1 + \frac{b^2}{64}} \]

### Step 5: Set up the equation

Since we found that \( e_{hyperbola} = 2 \), we set up the equation:

\[ 2 = \sqrt{1 + \frac{b^2}{64}} \]

### Step 6: Square both sides

Squaring both sides gives:

\[ 4 = 1 + \frac{b^2}{64} \]

### Step 7: Solve for \( b^2 \)

Subtracting 1 from both sides:

\[ 3 = \frac{b^2}{64} \]

Multiplying both sides by 64:

\[ b^2 = 192 \]

### Conclusion

Thus, the value of \( b^2 \) is:

\[ \boxed{192} \]