the length of the latusrectum of the ellipse `5x^(2) + 9x^(2)=45,` is

Options

A. `5//3`
B. `10//3`
C. `2sqrt(5)//5`
D. `sqrt(5)//3`

Answer

To solve the problem of finding the length of the latus rectum of the ellipse given by the equation \(5x^2 + 9y^2 = 45\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
First, we need to rewrite the given equation of the ellipse in standard form. The given equation is:

\[ 5x^2 + 9y^2 = 45 \]

To convert this to standard form, we divide the entire equation by 45:

\[ \frac{5x^2}{45} + \frac{9y^2}{45} = 1 \]

This simplifies to:

\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:

\[ a^2 = 9 \quad \text{and} \quad b^2 = 5 \]

### Step 3: Determine \(a\) and \(b\)
Now we calculate \(a\) and \(b\):

\[ a = \sqrt{9} = 3 \quad \text{and} \quad b = \sqrt{5} \]

### Step 4: Use the formula for the length of the latus rectum
For an ellipse where \(a > b\), the length of the latus rectum \(L\) is given by the formula:

\[ L = \frac{2b^2}{a} \]

### Step 5: Substitute the values of \(b^2\) and \(a\)
Now we substitute \(b^2 = 5\) and \(a = 3\) into the formula:

\[ L = \frac{2 \cdot 5}{3} = \frac{10}{3} \]

### Final Answer
Thus, the length of the latus rectum of the ellipse is:

\[ \frac{10}{3} \]

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