Q.If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

What is the Latus Rectum? A First Look

Imagine you draw a parabola, ellipse, or hyperbola — a conic section. Each of these curves has a special line segment that cuts straight across it, passing through the focus and running parallel to the directrix. That segment is the latus rectum.

The name comes from Latin: latus means "side" and rectum means "straight." So it's literally the "straight side" of the curve. For a student meeting it for the first time, think of it as the width of the curve at the focus — how wide the opening is right at that special point.

Intuition First

Take a parabola that opens upward, like $y^2 = 4ax$. Its focus is at $(a, 0)$. If you draw a vertical line through that focus, it will hit the parabola at two points — one above, one below. The distance between those two intersection points is the length of the latus rectum.

Why does this matter? Because it tells you how "fat" or "skinny" the curve is. A larger latus rectum means a wider opening; a smaller one means a tighter, narrower curve. For ellipses and hyperbolas, it also relates directly to the shape's eccentricity.

The Precise Statement

But that's just the parabola. Here are the exact formulas for all three conics:

Notice something interesting: the ellipse and hyperbola share the same formula. That's because both have two foci and the latus rectum is defined through either focus.

Why That Formula? A Quick Derivation

For the parabola $y^2 = 4ax$, the focus is at $(a, 0)$. A line through the focus parallel to the directrix is vertical (since the directrix is vertical $x = -a$). So the line is $x = a$.

Substitute $x = a$ into $y^2 = 4ax$:

The two intersection points are $(a, 2a)$ and $(a, -2a)$. The distance between them is $4a$. That's it.

Common Mistake to Avoid

Also, for ellipses and hyperbolas, there are two latera recta — one through each focus. They have the same length.

Why It Matters in Exams

You will be asked to:

• Find the length of the latus rectum given the equation of a conic.
• Use it to find $a$ or $b$ when the length is given.
• Relate it to eccentricity (especially for ellipses: $e = \sqrt{1 - \frac{b^2}{a^2}}$, so the latus rectum $\frac{2b^2}{a}$ ties directly to $e$).

One Final Intuition

Picture a parabola like a U-shaped slide. The latus rectum is the width of that slide exactly at the level of the focus. For an ellipse, imagine an oval — the latus rectum is the length of the vertical line segment that passes through one focus and touches the ellipse at two points. It's a concrete, measurable property that captures how "stretched" or "squashed" the curve is at its focal point.