Question

The area of the triangle formed by the lines joining the vertex of the parabola \(x^2 = 48y\) to the ends of its latus rectum, is .....

• A. \(144\) sq. units
• B. \(288\) sq. units
• C. \(72\) sq. units
• D. \(576\) sq. units

Hint:

Find a from 4a = 48, then the ends of the latus rectum are (plus or minus 2a, a), and use the triangle area.

Answer 1

The Correct Option is B

Step 1: Use the determinant formula.
Vertices: $(0,0)$, $(24,12)$, $(-24,12)$.
\[ \text{Area} = \tfrac{1}{2}\left|24\cdot 12 - (-24)\cdot 12\right| = \tfrac{1}{2}(288 + 288) = 288 \]

Step 2: Where did the points come from.
At $y = a = 12$, $x^2 = 48 \times 12 = 576$, so $x = \pm 24$.

Step 3: Compare.
Option (A) 144 is half of the area, and (D) 576 is double. Only (B) is correct.

Final Answer:
Option (B). \[ \boxed{288\text{ sq. units}} \]