Focus Directrix and Latus Rectum of a Parabola: Exam Tips

For the parabola y² = 4ax, the focus is the fixed point (a, 0), the directrix is the fixed line x = −a, and the latus rectum is the chord through the focus perpendicular to the axis, with endpoints (a, 2a) and (a, −2a) and total length 4a. Together these three objects completely pin down the parabola's size, position and orientation.

They matter because the focus–directrix definition is how the parabola is born: every point is equidistant from the two. The latus rectum is the natural "width" of the parabola at the focus and a favourite exam quantity — its endpoints give ready-made points satisfying the definition, and its equation (x = a for y² = 4ax) is a one-line answer once the form is recognised.

Key Concepts

• Focus: the fixed point in the parabola's definition; for y² = 4ax it is (a, 0), a units from the vertex along the axis.
• Directrix: the fixed line; for y² = 4ax it is x = −a, perpendicular to the axis and a units behind the vertex.
• Latus rectum: focal chord perpendicular to the axis; length 4a; endpoints (a, ±2a) for y² = 4ax.
• Eccentricity: every parabola has eccentricity e = 1, the ratio of distances to focus and directrix.

Formulas

• y² = 4ax: focus (a, 0); directrix x = −a; latus rectum x = a, length 4a, endpoints (a, 2a), (a, −2a).
• y² = −4ax: focus (−a, 0); directrix x = a; latus rectum x = −a, endpoints (−a, ±2a).
• x² = 4ay: focus (0, a); directrix y = −a; latus rectum y = a, endpoints (±2a, a).
• x² = −4ay: focus (0, −a); directrix y = a; latus rectum y = −a, endpoints (±2a, −a).
• Units: a in length units; focal distance vertex-to-focus equals vertex-to-directrix distance.

Worked Example

For y² = 8x, find the focus, directrix, latus rectum (equation, length, endpoints) and verify the definition at one endpoint.

1. Match y² = 4ax: 4a = 8, so a = 2. The parabola opens right.
2. Focus: (a, 0) = (2, 0). Directrix: x = −a, i.e. x = −2.
3. Latus rectum: the line x = 2; length 4a = 8; endpoints (2, 4) and (2, −4).
4. Verify with (2, 4): distance to focus = sqrt((2 − 2)² + (4 − 0)²) = 4; distance to directrix = |2 − (−2)| = 4. Equal, confirming the point is on the parabola.

Common Misconceptions

• "The latus rectum is 2a long." The semi-latus rectum is 2a; the full chord is 4a.
• "The directrix passes through the focus." It is perpendicular to the axis and 2a away from the focus, on the opposite side of the vertex.
• "Focus coordinates are (4a, 0)." The 4 appears in the equation's coefficient, but the focus is at distance a, not 4a, from the vertex.

Real-World Applications

• Dish antennas: the receiver is mounted at the focus; the latus rectum width helps size the feed assembly.
• Car headlights: the bulb sits at the focus so the reflector throws a parallel beam.
• Telescope mirrors: parabolic primary mirrors bring starlight to a focus with no spherical aberration.
• Microphone design: parabolic reflector microphones place the mic element at the focus to pick up distant sound.

In short: focus (a, 0), directrix x = −a, latus rectum x = a of length 4a — for y² = 4ax these three are read straight from a. Learn the pattern once and rotate it for the other three forms, then self-check with the equidistance property at a latus rectum endpoint.