The Parabola
A parabola is the locus of points equidistant from a fixed focus and a fixed directrix, with standard form y2 = 4ax.
What is The Parabola?
A parabola is the locus of points equidistant from a fixed focus and a fixed directrix, with standard form y2 = 4ax.
Key formula / rule: Focus is (a, 0) and directrix x = -a for y2 = 4ax.
Key points
- Derive the standard equation of a parabola from the focus-directrix definition.
- Compute focus, directrix and latus rectum for a given parabola.
Common exam trap
Swapping the roles of x and y between the two standard orientations.
Definitions
- Term
The Parabola
- Meaning
A parabola is the locus of points equidistant from a fixed focus and a fixed directrix, with standard form y2 = 4ax.
- Term
The Parabola — explanation
- Meaning
Every parabola property follows from the focus-directrix definition, including the reflective property that sends parallel rays to the focus. Latus rectum length 4a fixes the scale.
Learning objectives
Derive the standard equation of a parabola from the focus-directrix definition.
Compute focus, directrix and latus rectum for a given parabola.
Formulae
- Key point
Focus is (a, 0) and directrix x = -a for y2 = 4ax.
- Key point
Latus rectum has length 4a.
- Key point
Parametric point is (at2, 2at).
Prerequisites
MAT-U3-SL-T1-S2-C1
Common mistakes
Swapping the roles of x and y between the two standard orientations.
Taking the vertex as the focus when the parabola is shifted.
Keywords
Parabola
Practice preview
Which of the following equations represents a parabola with its vertex at the origin and axis along the x-axis?…
easy
A point P moves such that its distance from the point (2, 0) is equal to its distance from the line x = -2. The locus of P is a parabola. What is its equation?…
easy
For the parabola (y - 3)^2 = 16(x + 1), what are the coordinates of its vertex and focus?…
medium