Question

The area of the triangle formed by the vertex and the endpoints of the latus rectum of the parabola \(x^{2}=-18y\) is ...

• A. \(\frac{81}{4}\) square units.
• B. \(\frac{81}{2}\) square units
• C. \(\frac{81}{3}\) square units
• D. \(81\) square units

Hint:

The latus rectum has length 4a and the vertex is at distance a from it.

Answer 1

The Correct Option is B

Step 1: Endpoints
The focus is $\left(0,-\dfrac92\right)$.
Putting $y=-\dfrac92$: $x^{2}=81$, so $x=\pm9$.

Step 2: Triangle Vertices
The vertices are $(0,0)$, $\left(9,-\dfrac92\right)$ and $\left(-9,-\dfrac92\right)$.

Step 3: Area
Base $=18$, height $=\dfrac92$, area $=\dfrac12\cdot18\cdot\dfrac92=\dfrac{81}{2}$, option (B).

Final Answer:
The triangle with base 18 and height 9 over 2 has area 81 over 2. \[ \boxed{\text{(B) }\dfrac{81}{2}\text{ square units}} \]