If B and B' are the ends of minor axis and S and S' are the foci of the ellipse `x^(2)/25 + y^(2)/9 = 1`, then area of the rhombus SBS' B', in square units, will be

Options

A. 12
B. 48
C. 24
D. 36

Answer

To find the area of the rhombus formed by the points \( S, B, S', B' \) for the ellipse given by the equation

\[ \frac{x^2}{25} + \frac{y^2}{9} = 1, \]

we will follow these steps:

### Step 1: Identify the values of \( a \) and \( b \)

The standard form of an ellipse is

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

From the given equation, we can see that:

- \( a^2 = 25 \) → \( a = \sqrt{25} = 5 \)
- \( b^2 = 9 \) → \( b = \sqrt{9} = 3 \)

### Step 2: Determine the coordinates of the foci and ends of the minor axis

The foci \( S \) and \( S' \) of the ellipse are located at \( (\pm c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

Calculating \( c \):

\[ c = \sqrt{25 - 9} = \sqrt{16} = 4. \]

Thus, the coordinates of the foci are:

- \( S = (4, 0) \)
- \( S' = (-4, 0) \)

The ends of the minor axis \( B \) and \( B' \) are located at \( (0, \pm b) \):

- \( B = (0, 3) \)
- \( B' = (0, -3) \)

### Step 3: Calculate the area of the rhombus

The area \( A \) of a rhombus can be calculated using the formula:

\[ A = \frac{1}{2} \times d_1 \times d_2, \]

where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

In our case, the diagonals are:

- \( d_1 = BB' = |3 - (-3)| = 6 \)
- \( d_2 = SS' = |4 - (-4)| = 8 \)

Now substituting the values into the area formula:

\[ A = \frac{1}{2} \times 6 \times 8 = \frac{48}{2} = 24 \text{ square units.} \]

### Final Answer

Thus, the area of the rhombus \( SBS'B' \) is

\[ \boxed{24} \text{ square units.} \]

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