Question

If \( P(10, 2\sqrt{15}) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and the length of the latus rectum is 8, then the square of the area of \( \Delta PS_1S_2 \) is [where \( S_1 \) and \( S_2 \) are the foci of the hyperbola].

• A. 2700
• B. 2400
• C. 1750
• D. 3600

Hint:

In problems involving hyperbolas, the length of the latus rectum and the coordinates of points on the hyperbola are key to solving for unknowns.

Answer 1

The Correct Option is A

Step 1: Use the properties of hyperbolas.
The length of the latus rectum of a hyperbola is given by: \[ \text{Length of latus rectum} = \frac{2b^2}{a}. \] We are given that the length of the latus rectum is 8, so: \[ \frac{2b^2}{a} = 8. \] This gives the relation between \( a \) and \( b \). Step 2: Use the coordinates of point \( P(10, 2\sqrt{15}) \).
Since the point lies on the hyperbola, substitute \( x = 10 \) and \( y = 2\sqrt{15} \) into the equation of the hyperbola: \[ \frac{10^2}{a^2} - \frac{(2\sqrt{15})^2}{b^2} = 1. \] This gives another equation involving \( a \) and \( b \). Step 3: Solve the system of equations.
By solving these equations, we find the values for \( a \) and \( b \), and then calculate the square of the area of \( \Delta PS_1S_2 \). Final Answer: \[ \boxed{2700}. \]