Question

The area of the quadrilateral formed with the foci of the hyperbola \[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \] and its conjugate hyperbola is (in square units):

• A. \( 24 \)
• B. \( 16 \)
• C. \( 25 \)
• D. \( 50 \)

Hint:

For hyperbola problems, identify the foci using the formula \( c = \sqrt{a^2 + b^2} \), and use geometric properties to find areas.

Answer 1

The Correct Option is D

Step 1: Find the foci of the given hyperbola For a hyperbola of the form: \[ \frac{x^2}{a^2}
- \frac{y^2}{b^2} = 1, \] the foci are at: \[ (\pm c, 0), \quad \text{where } c = \sqrt{a^2 + b^2}. \] Given: \[ a^2 = 16, \quad b^2 = 9. \] \[ c = \sqrt{16 + 9} = \sqrt{25} = 5. \] Thus, the foci of the given hyperbola are at \( (\pm 5, 0) \). Step 2: Find the foci of the conjugate hyperbola For the conjugate hyperbola: \[ \frac{y^2}{16}
- \frac{x^2}{9} = 1, \] the foci are at \( (0, \pm c) \), where \( c \) is the same as above: \[ c = 5. \] Thus, the foci are at \( (0, \pm 5) \). Step 3: Compute the area of the quadrilateral The quadrilateral is a rectangle with sides: \[ \text{Length} = 10, \quad \text{Width} = 5. \] \[ \text{Area} = 10 \times 5 = 50. \]