The equation of the ellipse, the co-ordinates of whose foci a re `(pmsqrt(3), 0)` and length of the semi-major axis as 2 is

Options

A. `x^(2) + 4y^(2) =1 `
B. `4x^(2) + y^(2) + 4`
C. `4x^(2) + y^(2) =16`
D. `x^(2) + 4y^(2) = 4`

Answer

To find the equation of the ellipse given the coordinates of its foci and the length of the semi-major axis, we can follow these steps:

### Step 1: Identify the given information
- The coordinates of the foci are \((\pm p \sqrt{3}, 0)\).
- The length of the semi-major axis \(a = 2\).

### Step 2: Determine the value of \(c\)
The distance from the center of the ellipse to each focus is denoted by \(c\). From the given foci coordinates, we can see that:
\[ c = p \sqrt{3} \]

### Step 3: Use the relationship between \(a\), \(b\), and \(c\)
For an ellipse, the relationship between the semi-major axis \(a\), semi-minor axis \(b\), and the distance to the foci \(c\) is given by:
\[ c^2 = a^2 - b^2 \]

### Step 4: Substitute known values
We know:
- \(a = 2\) so \(a^2 = 2^2 = 4\)
- \(c = p \sqrt{3}\) so \(c^2 = (p \sqrt{3})^2 = 3p^2\)

Substituting these into the relationship gives:
\[ 3p^2 = 4 - b^2 \]

### Step 5: Solve for \(b^2\)
Rearranging the equation:
\[ b^2 = 4 - 3p^2 \]

### Step 6: Write the standard form of the ellipse
The standard form of the equation of an ellipse centered at the origin is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Substituting \(a^2\) and \(b^2\) into the equation:
\[ \frac{x^2}{4} + \frac{y^2}{4 - 3p^2} = 1 \]

### Step 7: Final equation of the ellipse
Thus, the equation of the ellipse is:
\[ \frac{x^2}{4} + \frac{y^2}{4 - 3p^2} = 1 \]