if the line x- 2y = 12 is tangent to the ellipse `(x^(2))/(b^(2))+(y^(2))/(b^(2))=1` at the point `(3,(-9)/(2))` then the length of the latusrectum of the ellipse is

Options

A. 9
B. `8sqrt(3)`
C. 5
D. `12sqrt(2)`

Explanation

To solve the problem, we need to find the length of the latus rectum of the ellipse given that the line \( x - 2y = 12 \) is tangent to the ellipse at the point \( (3, -\frac{9}{2}) \).

### Step-by-Step Solution:

1. **Identify the Equation of the Ellipse:**
The equation of the ellipse is given as:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

2. **Substitute the Tangent Point into the Ellipse Equation:**
We substitute the point \( (3, -\frac{9}{2}) \) into the ellipse equation:
\[ \frac{3^2}{a^2} + \frac{\left(-\frac{9}{2}\right)^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{9}{a^2} + \frac{\frac{81}{4}}{b^2} = 1 \]

3. **Write the Equation of the Tangent Line:**
The equation of the tangent line is given as:
\[ x - 2y = 12 \]
We can rearrange this to:
\[ x = 2y + 12 \]

4. **Use the Tangent Condition:**
The equation of the tangent to the ellipse at the point \( (x_1, y_1) \) is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Substituting \( x_1 = 3 \) and \( y_1 = -\frac{9}{2} \):
\[ \frac{3x}{a^2} + \frac{-\frac{9}{2}y}{b^2} = 1 \]

5. **Equate the Tangent Equations:**
We need to equate the two tangent equations:
\[ \frac{3x}{a^2} - \frac{9y}{2b^2} = 1 \quad \text{and} \quad x - 2y = 12 \]
Rearranging the second equation gives:
\[ x = 12 + 2y \]

6. **Substituting for \( x \):**
Substitute \( x = 12 + 2y \) into the tangent equation:
\[ \frac{3(12 + 2y)}{a^2} - \frac{9y}{2b^2} = 1 \]
Expanding gives:
\[ \frac{36 + 6y}{a^2} - \frac{9y}{2b^2} = 1 \]

7. **Separate the Terms:**
This leads to two equations:
\[ \frac{36}{a^2} = 1 \quad \Rightarrow \quad a^2 = 36 \]
and
\[ \frac{6y}{a^2} - \frac{9y}{2b^2} = 0 \]

8. **Solving for \( b^2 \):**
From the second equation, we can isolate \( b^2 \):
\[ 6y \cdot 2b^2 = 9y \cdot a^2 \]
Substituting \( a^2 = 36 \):
\[ 12b^2 = 9 \cdot 36 \quad \Rightarrow \quad b^2 = 27 \]

9. **Calculate the Length of the Latus Rectum:**
The length of the latus rectum \( L \) of the ellipse is given by:
\[ L = \frac{2b^2}{a} = \frac{2 \cdot 27}{6} = 9 \]

### Final Answer:
The length of the latus rectum of the ellipse is \( 9 \).