The eccentricity of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` be reciprocal to that of the ellipse `x^(2)+4y^(2)=4`. If the hyperbola passes through a focus of the ellipse, then :

Options

A. The equation of the hyperbola is `(x^(2))/(3)-(y^(2))/(2)=1`
B. a focus of the hyperbola is `(sqrt(3),0)`
C. the eccentricity of the hyperbola is `sqrt((5)/(3))`
D. the equation of the hyperbola is `x^(2)-3y^(2)=3`

Explanation

To solve the problem, we need to find the eccentricity of the hyperbola and relate it to the eccentricity of the given ellipse. We also need to ensure that the hyperbola passes through a focus of the ellipse.

### Step 1: Find the eccentricity of the ellipse

The equation of the ellipse is given as:

\[ x^2 + 4y^2 = 4 \]

We can rewrite this in standard form:

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

Here, \(a^2 = 4\) and \(b^2 = 1\). Therefore, \(a = 2\) and \(b = 1\).

The eccentricity \(e_2\) of the ellipse is given by:

\[ e_2 = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 2: Find the eccentricity of the hyperbola

The eccentricity \(e_1\) of the hyperbola given by:

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

is given by:

\[ e_1 = \sqrt{1 + \frac{b^2}{a^2}} \]

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

\[ e_1 = \frac{1}{e_2} = \frac{2}{\sqrt{3}} \]

### Step 3: Relate the eccentricities

Setting the expression for \(e_1\) equal to \(\frac{2}{\sqrt{3}}\):

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

Squaring both sides gives:

\[ 1 + \frac{b^2}{a^2} = \frac{4}{3} \]

Thus,

\[ \frac{b^2}{a^2} = \frac{4}{3} - 1 = \frac{1}{3} \]

This implies:

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

### Step 4: Find the coordinates of the focus of the ellipse

The foci of the ellipse are located at:

\[ (\pm ae_2, 0) = \left(\pm 2 \cdot \frac{\sqrt{3}}{2}, 0\right) = (\pm \sqrt{3}, 0) \]

### Step 5: Substitute the focus into the hyperbola equation

The hyperbola passes through the focus of the ellipse, which is \((\sqrt{3}, 0)\). Substituting this into the hyperbola equation:

\[ \frac{(\sqrt{3})^2}{a^2} - \frac{0^2}{b^2} = 1 \]

This simplifies to:

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

### Step 6: Find \(b^2\)

Using \(b^2 = \frac{1}{3} a^2\):

\[ b^2 = \frac{1}{3} \cdot 3 = 1 \]

### Step 7: Write the equation of the hyperbola

Now we can write the equation of the hyperbola:

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

This can be rearranged to:

\[ x^2 - 3y^2 = 3 \]

### Final Result

Thus, the equation of the hyperbola is:

\[ x^2 - 3y^2 = 3 \]