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 \( S \) and \( S' \) to any tangent of the ellipse given by the equation:

\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]

### Step 1: Identify the parameters of the ellipse
The given equation of the ellipse can be rewritten in the standard form:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Here, we have \( a^2 = 4 \) and \( b^2 = 9 \). Thus, \( a = 2 \) and \( b = 3 \).

### Step 2: Determine the foci of the ellipse
The foci of the ellipse can be found using the formula:

\[ c = \sqrt{b^2 - a^2} \]

Substituting the values of \( a \) and \( b \):

\[ c = \sqrt{9 - 4} = \sqrt{5} \]

### Step 3: Use the property of the ellipse
For any tangent to the ellipse, the lengths of the perpendiculars from the foci \( S \) and \( S' \) to the tangent are related by the equation:

\[ ac = b^2 \]

### Step 4: Substitute the values
We already know that \( b^2 = 9 \). Therefore, we can write:

\[ ac = 9 \]

### Conclusion
Thus, the value of \( ac \) is equal to \( 9 \).

### Final Answer:
\[ \boxed{9} \]