The length of perpendicular from the foci S and S on any tangent to ellipse `(x^(2))/(4)+(y^(2))/(9)=1` are a and c respectively then the value of `ac` is equal to

Options

A. 4
B. 18
C. 8
D. none of these

Answer

To solve the problem, we need to find the value of \( ac \) where \( a \) and \( c \) are the lengths of the perpendiculars from the foci of the ellipse to any tangent line.

### Step-by-Step Solution:

1. **Identify the given ellipse**:
The equation of the ellipse is given as:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
Here, we can identify \( a^2 = 4 \) and \( b^2 = 9 \).

2. **Determine the values of \( a \) and \( b \)**:
From \( a^2 = 4 \) and \( b^2 = 9 \):
\[ a = \sqrt{4} = 2, \quad b = \sqrt{9} = 3 \]

3. **Find the foci of the ellipse**:
The foci \( S \) and \( S' \) of the ellipse are located at:
\[ c = \sqrt{b^2 - a^2} = \sqrt{9 - 4} = \sqrt{5} \]
Therefore, the foci are at \( (0, \pm \sqrt{5}) \).

4. **Use the property of the ellipse**:
According to the property of the ellipse, the product of the lengths of the perpendiculars from the foci to any tangent is given by:
\[ ac = b^2 \quad \text{(since \( b^2 > a^2 \))} \]

5. **Calculate \( b^2 \)**:
We already found \( b^2 = 9 \).

6. **Conclusion**:
Therefore, the value of \( ac \) is:
\[ ac = 9 \]

### Final Answer:
The value of \( ac \) is \( 9 \).