Question

The area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 20y$ to the end of its latus rectum is

• A. $100$ sq. units
• B. $40$ sq. units
• C. $20$ sq. units
• D. $50$ sq. units

Hint:

The area of the triangle formed by the vertex and the latus rectum is always $2a^2$.

Answer 1

The Correct Option is D

Step 1: Concept
For a parabola $x^2 = 4ay$, the vertex is $(0, 0)$ and the ends of the latus rectum are $(2a, a)$ and $(-2a, a)$.

Step 2: Meaning
From $x^2 = 20y$, we have $4a = 20 \implies a = 5$.

Step 3: Analysis
Vertex $V = (0, 0)$. Ends of latus rectum: $L_1 = (10, 5)$ and $L_2 = (-10, 5)$. The triangle has a base (the latus rectum) of length $4a = 20$. The height of the triangle is the distance from the vertex to the latus rectum, which is $a = 5$. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 5 = 50$.

Step 4: Conclusion
The area of the triangle is 50 sq. units. Final Answer: (D)