What is the length of the latus rectum of the parabola having focus at $(0, 4)$ and directrix $y = - 4$ ?

hard 3 marks Class 11 · Mathematics · Conic Sections

Answer

The correct answer is 16 — Latus rectum of a parabola is $4 a$ , thus $4 \times 4 = 16$ .

Step-by-step solution

Find the vertex as the midpoint of focus and directrix: $\frac{4 + ( - 4 )}{2} = 0$ , so vertex is $(0, 0)$ .
Use distance from vertex to focus $a = 4$ as the focal length.
Write the standard form of a vertical parabola: $x^{2} = 4 a y$ with $a = 4$ to get $x^{2} = 16 y$ .
Latus rectum of a parabola is $4 a$ , thus $4 \times 4 = 16$ .

Common mistake: Confusing the axis direction or using

2 a

instead of

4 a

Concept tested: Parabola – latus rectum