The Latus Rectum of a conic section is a chord that passes through the focus and is perpendicular to the principal axis. For a parabola, the latus rectum has a specific, easily calculated length.
The latus rectum is important in optics — in a parabolic mirror or lens, rays that pass through the endpoints of the latus rectum reflect parallel to the axis. This property is used in parabolic satellite dishes and reflectors.
For the standard form of a parabola y² = 4ax (opening to the right, focus at (a, 0)):
Derivation: The focus is at F(a, 0). The latus rectum is the vertical chord through F. Substituting x = a in y² = 4ax: y² = 4a² → y = ±2a So the endpoints are (a, 2a) and (a, -2a) Length = 2a - (-2a) = 4a ✓
| Parabola | Focus | Length of LR |
|---|---|---|
| y² = 4ax | (a, 0) | 4a |
| y² = -4ax | (-a, 0) | 4a |
| x² = 4ay | (0, a) | 4a |
| x² = -4ay | (0, -a) | 4a |
In all standard forms, the length of the latus rectum = |4a| where 'a' is the distance from vertex to focus (the parameter of the parabola).
Comparing y² = 8x with y² = 4ax: 4a = 8, so a = 2. Length of latus rectum = 4a = **8 units**. The focus is at (2, 0).
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