Find the ellipse if its foci are `(pm2, 0)` and the length of the latus rectum is `10/3`.

Answer

To find the equation of the ellipse given its foci and the length of the latus rectum, we can follow these steps:

### Step 1: Identify the given information
- The foci of the ellipse are at points \((\pm 2, 0)\).
- The length of the latus rectum is \(\frac{10}{3}\).

### Step 2: Determine the center and orientation of the ellipse
Since the foci are located on the x-axis, we can conclude that the center of the ellipse is at the origin \((0, 0)\) and the major axis is horizontal.

### Step 3: Use the relationship for foci
The distance from the center to each focus is denoted as \(c\). Given the foci \((\pm 2, 0)\), we have:
\[ c = 2 \]
Thus,
\[ c^2 = 4 \]

### Step 4: Use the formula for the length of the latus rectum
The length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2b^2}{a} \]
We know that \(L = \frac{10}{3}\), so we can set up the equation:
\[ \frac{2b^2}{a} = \frac{10}{3} \]

### Step 5: Rearranging the equation
From the equation above, we can express \(b^2\) in terms of \(a\):
\[ 2b^2 = \frac{10}{3}a \]
\[ b^2 = \frac{5}{3}a \]

### Step 6: Use the relationship between \(a\), \(b\), and \(c\)
For an ellipse, we have the relationship:
\[ c^2 = a^2 - b^2 \]
Substituting the known values:
\[ 4 = a^2 - b^2 \]
Now substituting \(b^2\) from the previous step:
\[ 4 = a^2 - \frac{5}{3}a \]

### Step 7: Rearranging to form a quadratic equation
Multiplying through by 3 to eliminate the fraction:
\[ 12 = 3a^2 - 5a \]
Rearranging gives:
\[ 3a^2 - 5a - 12 = 0 \]

### Step 8: Factor the quadratic equation
To factor the quadratic equation:
\[ 3a^2 - 9a + 4a - 12 = 0 \]
Grouping the terms:
\[ 3a(a - 3) + 4(a - 3) = 0 \]
Factoring out \((a - 3)\):
\[ (3a + 4)(a - 3) = 0 \]

### Step 9: Solve for \(a\)
Setting each factor to zero gives:
1. \(3a + 4 = 0 \Rightarrow a = -\frac{4}{3}\) (not valid since \(a\) must be positive)
2. \(a - 3 = 0 \Rightarrow a = 3\)

### Step 10: Find \(b^2\)
Now substitute \(a = 3\) back into the equation for \(b^2\):
\[ b^2 = \frac{5}{3} \times 3 = 5 \]

### Step 11: Write the equation of the ellipse
The standard form of the ellipse with a horizontal major axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \(a^2 = 9\) and \(b^2 = 5\):
\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \]

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