Parabola

The defining property. Fix a point $F$ (the focus) and a line $\ell$ (the directrix) not through $F$. A parabola is the set of every point $P$ whose distance to $F$ equals its perpendicular distance to $\ell$. Nothing else about the curve is assumed — everything else, including its equation, is derived from this one balancing condition.

Setting up coordinates. Place the vertex at the origin with the axis along the positive $x$-axis, focus at $F(a,0)$ for $a>0$, and directrix $x=-a$. This is the natural choice because it makes the vertex — the point on the curve closest to both $F$ and $\ell$ — sit exactly midway between them, at distance $a$ from each.

The standard equation. Writing the defining property as a distance equation and simplifying (shown in full below) gives

where: $x,y$ are coordinates of a general point on the curve, and $a>0$ is the fixed distance from vertex to focus (and from vertex to directrix).

Reading off the features. For $y^2=4ax$: the vertex is $(0,0)$, the axis is the $x$-axis, the focus is $(a,0)$, and the directrix is $x=-a$. The latus rectum — the focal chord perpendicular to the axis — has length $4a$, running between $(a,2a)$ and $(a,-2a)$; substituting $x=a$ into $y^2=4ax$ gives $y^2=4a^2$, so $y=\pm2a$, confirming both the endpoints and the length $4a$.

The other three orientations. Reflecting or rotating the picture gives $y^2=-4ax$ (opens left, focus $(-a,0)$, directrix $x=a$), $x^2=4ay$ (opens upward, focus $(0,a)$, directrix $y=-a$), and $x^2=-4ay$ (opens downward, focus $(0,-a)$, directrix $y=a$). A quick way to keep them straight: the squared variable names the axis of symmetry, and the sign of the coefficient names the direction — positive for right or up, negative for left or down.

Why the vertex is special. Among all points on the curve, the vertex is the unique one lying exactly on the axis of symmetry, and it sits precisely midway between focus and directrix — a distance of $a$ from each. Geometrically it is the sharpest-turning point of the curve: as you move away from the vertex along the parabola in either direction, the curve bends more and more gently, opening ever wider, which is why the parabola never closes up into an oval the way an ellipse does. This single-turning-point structure is also what makes a parabola useful as a reflector shape: every ray parallel to the axis, after bouncing off the curve, is redirected to pass through the one fixed focus, which is exactly why headlight and satellite-dish reflectors are built to this curve rather than any other.

Find the focus, directrix, and length of the latus rectum of the parabola $y^2=12x$; then find the equation of a second parabola with vertex at the origin, axis along the $y$-axis, opening upward, that passes through $(6,3)$.

Step 1. Compare $y^2=12x$ with the standard form $y^2=4ax$: $4a=12$, so $a=3$. Reading off the features: focus $(a,0)=(3,0)$, directrix $x=-a=-3$, and length of latus rectum $4a=12$.

Step 2. For the second parabola, the axis is the $y$-axis and it opens upward with vertex at the origin, so its form is $x^2=4ay$ for some $a>0$ still to be found.

Step 3. Substitute the given point $(6,3)$ into $x^2=4ay$: $6^2=4a(3)$, so $36=12a$, giving $a=3$.

Step 4. Write the equation and check it. With $a=3$, the equation is $x^2=4(3)y=12y$. Substituting $(6,3)$ back: left side $6^2=36$, right side $12(3)=36$ — they match, confirming the equation.

Final answer: for $y^2=12x$, focus $(3,0)$, directrix $x=-3$, latus rectum $12$; the second parabola is $x^2=12y$.

Competitive stretch

A parabola $y^2=4ax$ has latus rectum length $20$. What is the distance from its focus to its directrix? Since $4a=20$, $a=5$, and the focus $(a,0)$ to the directrix $x=-a$ is a horizontal gap of $a-(-a)=2a=10$ — exactly twice $a$, and, notably, always equal to half the latus rectum's own length only when $4a=2\cdot 2a$, i.e. never in general; here the focus-to-directrix distance $2a=10$ happens to be less than the latus rectum $4a=20$, illustrating that these two lengths are related but not equal.

• Mixing up which coordinate is squared. Horizontal-axis parabolas have $y^2=4ax$ (the $y$-term squared); vertical-axis parabolas have $x^2=4ay$ (the $x$-term squared) — swapping these is the single most common slip.
• Misreading $a$ from the coefficient. Given $y^2=12x$, it is tempting to set $a=12$ directly; remember $4a=12$, so $a=3$, and the focus and directrix use this smaller value, not the raw coefficient.
• Getting the opening direction's sign wrong. A positive coefficient (as in $y^2=4ax$ or $x^2=4ay$) opens in the positive direction (right or up); a negative one (as in $y^2=-4ax$) opens in the negative direction (left or down).
• Assuming the vertex is always the origin. The standard forms above all assume vertex $(0,0)$; a parabola with vertex elsewhere needs a shift of origin first, translating $x\to x-h$, $y\to y-k$ for vertex $(h,k)$.

Ask the tutor: If the four standard orientations of a parabola keep blurring together, ask the AI tutor to quiz you rapid-fire on focus, directrix, and latus rectum for each one until the pattern is automatic.