The equation of ellipse whose foci are `(pm 3, 0)` and length of semi-major axis is 4 is

Options

A. A. `(x^(2))/(7)+(y^(2))/(16)=1`
B. B. `(x^(2))/(16)+(y^(2))/(7)=1`
C. C. `(x^(2))/(9)+(y^(2))/(25)=1`
D. D. `(x^(2))/(9)+(y^(2))/(25)=1`

Answer

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

### Step 1: Identify the given information
The foci of the ellipse are at points \((\pm 3, 0)\) and the length of the semi-major axis is \(4\).

### Step 2: Determine the value of \(a\)
The semi-major axis \(a\) is given as \(4\). Therefore, we can calculate \(a^2\):
\[ a^2 = 4^2 = 16 \]

### Step 3: Determine the distance of the foci from the center
The distance of the foci from the center of the ellipse is given as \(c = 3\). The relationship between \(a\), \(b\), and \(c\) in an ellipse is given by:
\[ c^2 = a^2 - b^2 \]

### Step 4: Calculate \(c^2\)
We can calculate \(c^2\):
\[ c^2 = 3^2 = 9 \]

### Step 5: Substitute \(a^2\) and \(c^2\) into the equation
Now, we substitute \(a^2\) and \(c^2\) into the relationship:
\[ 9 = 16 - b^2 \]

### Step 6: Solve for \(b^2\)
Rearranging the equation gives us:
\[ b^2 = 16 - 9 = 7 \]

### Step 7: Write the equation of the ellipse
The standard form of the equation of an ellipse with a horizontal major axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \(a^2\) and \(b^2\) into the equation, we have:
\[ \frac{x^2}{16} + \frac{y^2}{7} = 1 \]

### Conclusion
Thus, the equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{7} = 1 \]