The parabola

A parabola is the set of all points that are equally far from a fixed line (the directrix) and a fixed point (the focus) not on that line.

• The line through the focus perpendicular to the directrix is the axis; the parabola is symmetric about it.
• The point where the parabola meets its axis is the vertex, exactly half-way between the focus and the directrix.

With the vertex at the origin and the focus a distance a away (a > 0), there are four standard forms:

How to read one: the squared letter tells you the axis is the other axis (y² means symmetric about the x-axis); the sign of the other side tells you the direction it opens.

• y² = 8x: 4a = 8, so a = 2. Opens right, focus (2, 0), directrix x = –2.
• x² = –12y: 4a = 12, so a = 3. Opens down, focus (0, –3), directrix y = 3.

The latus rectum is the chord through the focus, perpendicular to the axis. For y² = 4ax its ends are (a, ±2a), so its length is 4a: 8 for y² = 8x, 12 for x² = –12y.

Finding the equation.

• Focus (0, 4) and directrix y = –4: the vertex is at the origin, the focus is above it, so x² = 4ay with a = 4: x² = 16y.
• Vertex at the origin, axis along the x-axis, passing through (1, 4): the point is to the right, so y² = 4ax, and 16 = 4a × 1 gives a = 4: y² = 16x.

Check any answer with the definition: on y² = 16x the point (1, 4) is 1 + 4 = 5 from the directrix x = –4, and √((1 – 4)² + 4²) = 5 from the focus (4, 0). Equal, as they should be.