Parabola
Standard Equations
A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed straight line. The fixed point is the focus and the fixed line is the directrix. This balance between a point and a line gives the parabola its single, sweeping branch.
Some vocabulary you will use throughout: the axis is the line through the focus perpendicular to the directrix (the curve's line of symmetry); the vertex is the point where the parabola meets its axis, sitting exactly midway between focus and directrix; and the latus rectum is the chord through the focus perpendicular to the axis. For all four standard forms below its length is the same:
Placing the vertex at the origin and the axis along a coordinate axis gives the four standard forms. The sign and which variable is squared tell you instantly which way the parabola opens:
| Equation | Opens | Focus | Directrix | Axis |
|---|---|---|---|---|
| $y^2 = 4ax$ | right | $(a, 0)$ | $x = -a$ | $y = 0$ |
| $y^2 = -4ax$ | left | $(-a, 0)$ | $x = a$ | $y = 0$ |
| $x^2 = 4ay$ | upward | $(0, a)$ | $y = -a$ | $x = 0$ |
| $x^2 = -4ay$ | downward | $(0, -a)$ | $y = a$ | $x = 0$ |
In every case $a > 0$ is the distance from the vertex to the focus (and equally from the vertex to the directrix). A clean way to remember the orientation: if $y$ is squared the parabola opens horizontally, if $x$ is squared it opens vertically; the sign then chooses the direction.
Deeper Insight — why "equal distances" forces a curve, and where it shows up: The defining condition $PF = PM$ (distance to focus equals perpendicular distance to directrix) is deceptively simple, yet squaring it produces exactly a second-degree equation with only one squared variable — which is what makes a parabola distinct from a circle or an ellipse. The constant $4a$ is not decorative: it is forced by the algebra of equating those two distances, and it is the reason the latus rectum length is always $|4a|$. This focus-directrix balance is also why parabolas are everywhere in the physical world — the path of a projectile under gravity, the cross-section of a satellite dish, the reflector behind a torch bulb. A parabolic mirror gathers every ray parallel to the axis to the single focus, a property that falls straight out of the equal-distance definition. Master the four standard forms not as four separate facts but as one idea seen from four directions: square one variable, set it equal to $4a$ times the other, and let the sign point the way.
- Compare with $y^2 = 4ax$: so $4a = 12$, giving $a = 3$.
- The parabola opens right, so focus is $(a, 0) = (3, 0)$.
- Directrix is $x = -a$, i.e. $x = -3$.
- Length of latus rectum $= 4a = 12$.
Answer: Focus $(3, 0)$, directrix $x = -3$, latus rectum $12$.
- Axis along $y$-axis and opening upward means the form $x^2 = 4ay$.
- Substitute the point $(4, 2)$: $4^2 = 4a(2) \Rightarrow 16 = 8a \Rightarrow a = 2$.
- So $4a = 8$ and the equation is $x^2 = 8y$.
Answer: $x^2 = 8y$.
- The focus is below the origin and the directrix is above it, so the parabola opens downward: $x^2 = -4ay$.
- The focus is $(0, -a)$, so $-a = -3 \Rightarrow a = 3$.
- Thus $4a = 12$ and the equation is $x^2 = -12y$.
Answer: $x^2 = -12y$.
- Compare with $y^2 = -4ax$: so $4a = 8 \Rightarrow a = 2$.
- The negative sign with $y^2$ means it opens to the left.
- Focus of $y^2 = -4ax$ is $(-a, 0) = (-2, 0)$.
Answer: Opens left; focus $(-2, 0)$.
- The focus lies on the positive $x$-axis, so the parabola opens right: $y^2 = 4ax$.
- Focus is $(a, 0) = (5, 0)$, so $a = 5$.
- Then $4a = 20$, giving $y^2 = 20x$.
Answer: $y^2 = 20x$.
- Here $4a = 16 \Rightarrow a = 4$, so the focus is $(4, 0)$.
- The latus rectum is the vertical chord through the focus; substitute $x = 4$: $y^2 = 16(4) = 64 \Rightarrow y = \pm 8$.
- So the endpoints are $(4, 8)$ and $(4, -8)$ — a chord of length $16 = 4a$, as expected.
Answer: $(4, 8)$ and $(4, -8)$.
- A parabola is the set of points equidistant from a focus and a directrix.
- Four standard forms: $y^2 = \pm 4ax$ (opens horizontally), $x^2 = \pm 4ay$ (opens vertically).
- For $y^2 = 4ax$: focus $(a,0)$, directrix $x = -a$, axis $y = 0$, vertex at origin.
- The latus rectum has length $|4a|$ for all four forms.
- $y$ squared → opens left/right; $x$ squared → opens up/down; the sign sets the direction.
Focus, Directrix and Latus Rectum
For y² = 4ax: focus (a, 0), directrix x = −a, axis the x-axis, and latus rectum 4a.
- y² = 4ax: focus (a, 0), directrix x = −a.
- Latus rectum length = 4a.
Reading the Elements
Reduce a given equation to a standard form, identify a, then read off vertex, focus, directrix and latus rectum.
- Match the equation to a standard form to find a.
- Then read vertex, focus, directrix, latus rectum.