The length of the latusrectum of `9x^(2) + 25y^(2) - 90x - 150y + 225 = 0` is

Options

A. `9/5`
B. `18/5`
C. `18/25`
D. `9/25`

Answer

To find the length of the latus rectum of the given ellipse represented by the equation \(9x^2 + 25y^2 - 90x - 150y + 225 = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to group the \(x\) and \(y\) terms:
\[ 9x^2 - 90x + 25y^2 - 150y + 225 = 0 \]

### Step 2: Completing the Square for \(x\) Terms
For the \(x\) terms \(9x^2 - 90x\):
1. Factor out the coefficient of \(x^2\):
\[ 9(x^2 - 10x) \]
2. Complete the square:
\[ x^2 - 10x = (x - 5)^2 - 25 \]
Thus,
\[ 9((x - 5)^2 - 25) = 9(x - 5)^2 - 225 \]

### Step 3: Completing the Square for \(y\) Terms
For the \(y\) terms \(25y^2 - 150y\):
1. Factor out the coefficient of \(y^2\):
\[ 25(y^2 - 6y) \]
2. Complete the square:
\[ y^2 - 6y = (y - 3)^2 - 9 \]
Thus,
\[ 25((y - 3)^2 - 9) = 25(y - 3)^2 - 225 \]

### Step 4: Substitute Back into the Equation
Now substitute the completed squares back into the equation:
\[ 9((x - 5)^2 - 25) + 25((y - 3)^2 - 9) + 225 = 0 \]
This simplifies to:
\[ 9(x - 5)^2 - 225 + 25(y - 3)^2 - 225 + 225 = 0 \]
\[ 9(x - 5)^2 + 25(y - 3)^2 - 225 = 0 \]
\[ 9(x - 5)^2 + 25(y - 3)^2 = 225 \]

### Step 5: Dividing by 225
Divide the entire equation by 225 to get it into standard form:
\[ \frac{9(x - 5)^2}{225} + \frac{25(y - 3)^2}{225} = 1 \]
This simplifies to:
\[ \frac{(x - 5)^2}{25} + \frac{(y - 3)^2}{9} = 1 \]

### Step 6: Identifying Parameters
From the standard form of the ellipse:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
we identify:
- \(a^2 = 25 \Rightarrow a = 5\)
- \(b^2 = 9 \Rightarrow b = 3\)

### Step 7: Calculating the Length of the Latus Rectum
The length of the latus rectum \(L\) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b\) and \(a\):
\[ L = \frac{2 \cdot 9}{5} = \frac{18}{5} \]

### Final Answer
The length of the latus rectum is \(\frac{18}{5}\).

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