If the foci of `(x^(2))/(16)+(y^(2))/(4)=1` and `(x^(2))/(a^(2))-(y^(2))/(3)=1` coincide, the value of a is

Options

A. 3
B. 2
C. `(1)/(sqrt(3))`
D. `sqrt(3)`

Explanation

To solve the problem, we need to find the value of \( a \) such that the foci of the given ellipse and hyperbola coincide. Let's go through the solution step by step.

### Step 1: Identify the parameters of the ellipse
The equation of the ellipse is given by:

\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]

From this equation, we can identify:
- \( a^2 = 16 \) (where \( a = 4 \))
- \( b^2 = 4 \) (where \( b = 2 \))

### Step 2: Calculate the eccentricity of the ellipse
The eccentricity \( e \) of the ellipse is given by the formula:

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

Substituting the values of \( a^2 \) and \( b^2 \):

\[ e = \sqrt{1 - \frac{4}{16}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 3: Calculate the foci of the ellipse
The foci of the ellipse are given by:

\[ (\pm ae, 0) \]

Substituting \( a \) and \( e \):

\[ \text{Foci of ellipse} = \left( \pm 4 \cdot \frac{\sqrt{3}}{2}, 0 \right) = \left( \pm 2\sqrt{3}, 0 \right) \]

### Step 4: Identify the parameters of the hyperbola
The equation of the hyperbola is given by:

\[ \frac{x^2}{a^2} - \frac{y^2}{3} = 1 \]

From this equation, we can identify:
- \( b^2 = 3 \)

### Step 5: Calculate the eccentricity of the hyperbola
The eccentricity \( e \) of the hyperbola is given by the formula:

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

Substituting \( b^2 \):

\[ e = \sqrt{1 + \frac{3}{a^2}} \]

### Step 6: Calculate the foci of the hyperbola
The foci of the hyperbola are given by:

\[ (\pm ae, 0) \]

Substituting the expression for \( e \):

\[ \text{Foci of hyperbola} = \left( \pm a \sqrt{1 + \frac{3}{a^2}}, 0 \right) \]

### Step 7: Set the foci of the ellipse equal to the foci of the hyperbola
Since the foci coincide, we set:

\[ 2\sqrt{3} = a \sqrt{1 + \frac{3}{a^2}} \]

### Step 8: Square both sides to eliminate the square root
Squaring both sides gives:

\[ (2\sqrt{3})^2 = a^2 \left(1 + \frac{3}{a^2}\right) \]

This simplifies to:

\[ 12 = a^2 + 3 \]

### Step 9: Solve for \( a^2 \)
Rearranging gives:

\[ a^2 = 12 - 3 = 9 \]

### Step 10: Find \( a \)
Taking the square root:

\[ a = 3 \]

Thus, the value of \( a \) is:

\[ \boxed{3} \]