The length of the latusrectum of `((x-3)^(2))/(16)+(y-2)^(2)/(36)=1` is

Options

A. `8/3`
B. `8/9`
C. `32/3`
D. `16/3`

Answer

To find the length of the latus rectum of the given ellipse \(\frac{(x-3)^2}{16} + \frac{(y-2)^2}{36} = 1\), we can follow these steps:

### Step 1: Identify the values of \(a^2\) and \(b^2\)
The standard form of the ellipse is given by:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

where \((h, k)\) is the center of the ellipse, \(a^2\) is the denominator under the \(x\) term, and \(b^2\) is the denominator under the \(y\) term.

From the given equation:

\[ \frac{(x-3)^2}{16} + \frac{(y-2)^2}{36} = 1 \]

we can identify:

- \(a^2 = 16\)
- \(b^2 = 36\)

### Step 2: Determine \(a\) and \(b\)
Now we can find \(a\) and \(b\):

\[ a = \sqrt{16} = 4 \]
\[ b = \sqrt{36} = 6 \]

### Step 3: Identify the formula for the length of the latus rectum
Since \(b > a\) in this case (6 > 4), we use the formula for the length of the latus rectum:

\[ \text{Length of latus rectum} = \frac{2a^2}{b} \]

### Step 4: Substitute the values into the formula
Now we substitute the values of \(a^2\) and \(b\) into the formula:

\[ \text{Length of latus rectum} = \frac{2 \times 16}{6} \]

### Step 5: Simplify the expression
Now we simplify the expression:

\[ \text{Length of latus rectum} = \frac{32}{6} = \frac{16}{3} \]

### Final Answer
Thus, the length of the latus rectum is \(\frac{16}{3}\).

---