Find the equation of the ellipse whose major axis is 12 and foci are `(pm4,0)`.

Answer

To find the equation of the ellipse whose major axis is 12 and foci are at (±4, 0), we can follow these steps:

### Step 1: Identify the values of a and c
The length of the major axis is given as 12. The semi-major axis \( a \) is half of this length:
\[ a = \frac{12}{2} = 6 \]

The foci are located at (±4, 0), which gives us the value of \( c \):
\[ c = 4 \]

### Step 2: 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 3: Substitute the known values
We already have \( a = 6 \) and \( c = 4 \). Now we can substitute these values into the equation:
\[ 4^2 = 6^2 - b^2 \]
\[ 16 = 36 - b^2 \]

### Step 4: Solve for b^2
Rearranging the equation gives us:
\[ b^2 = 36 - 16 \]
\[ b^2 = 20 \]

### Step 5: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin with the major axis along the x-axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 = 36 \) and \( b^2 = 20 \) into the equation gives:
\[ \frac{x^2}{36} + \frac{y^2}{20} = 1 \]

### Final Answer
Thus, the equation of the ellipse is:
\[ \frac{x^2}{36} + \frac{y^2}{20} = 1 \]

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