Latus Rectum

Definition. For a parabola, the latus rectum is the line segment that passes through the focus, is perpendicular to the axis of the parabola, and has its endpoints on the parabola.

Intuition. Think of the latus rectum as the "width" of the parabola measured right at the focus — it tells you how wide the curve opens at that single special point. The axis is the line of symmetry; the latus rectum cuts across it at a right angle, straddling the focus.

Length. For the standard parabola $y^2 = 4ax$ (opening to the right), the length of the latus rectum is $4a$. This comes from the definition: the distance from the focus to the parabola along the latus rectum is $2a$, and because the parabola is symmetric about the $x$-axis, the full segment is twice that, giving $4a$. …

Definition. For a parabola, the latus rectum is the line segment that passes through the focus, is perpendicular to the axis of the parabola, and has its endpoints on the parabola.

Intuition. Think of the latus rectum as the "width" of the parabola measured right at the focus — it tells you how wide the curve opens at that single special point. The axis is the line of symmetry; the latus rectum cuts across it at a right angle, straddling the focus.

Length. For the standard parabola $y^2 = 4ax$ (opening to the right), the length of the latus rectum is $4a$. This comes from the definition: the distance from the focus to the parabola along the latus rectum is $2a$, and because the parabola is symmetric about the $x$-axis, the full segment is twice that, giving $4a$. …

The figure shows a standard right‑opening parabola drawn on an ordinary $xy$‑plane. The vertex is at the origin $O$, and the axis of the parabola is the positive $x$‑axis. The focus $F$ lies on this axis, inside the curve. A blue line segment is drawn through the focus, perpendicular to the axis — that is, vertical — with its two endpoints touching the parabola. This segment is labelled Latus rectum.

The physical idea is simple: the latus rectum is the chord of the parabola that passes through the focus and is perpendicular to the axis. It is the widest chord you can draw through the focus, and its length turns out to be a clean, fixed multiple of the focal distance $a$. The figure makes this geometric definition concrete: you see the focus, the axis, and the perpendicular chord all in one picture.

From this diagram the textbook derives the length of the latus rectum for the standard parabola $y^2 = 4ax$. Here is the reasoning step by step.

Let the endpoints of the latus rectum be $A$ and $B$, with $A$ above the $x$‑axis and $B$ below. By definition of a parabola, every point on the curve is equidistant from the focus $F$ and the directrix. For point $A$, this means $AF = AC$, where $C$ is the foot of the perpendicular from $A$ to the directrix. But $AC$ is exactly the horizontal distance from $A$ to the directrix, which equals the distance $FM$ — the distance from the focus to the directrix. Since the focus is at $(a,0)$ and the directrix is $x = -a$, we have $FM = 2a$. Hence $AF = 2a$.

The parabola is symmetric about the $x$‑axis, so $AF = FB$. Therefore the full length $AB = AF + FB = 2a + 2a = 4a$.

Fig. 10.18 is a coordinate-plane diagram of the parabola $y^2 = 4ax$ (with $a > 0$). The x-axis is horizontal, the y-axis vertical. The parabola opens to the right, with its vertex at the origin $O(0,0)$. The axis of the parabola is the x-axis itself.

The focus is labelled $F(a,0)$ — a point on the x-axis inside the right-opening curve. The directrix is the vertical line $x = -a$, and the point where this line meets the x-axis is labelled $M(-a,0)$. The latus rectum is drawn as a vertical blue line segment through $F$, with endpoints on the parabola: $A(a,2a)$ above the x-axis and $B(a,-2a)$ below it. So the latus rectum runs from $A$ straight down to $B$, passing through $F$ at its midpoint.

The figure also shows a point $C(-a,2a)$ — directly above $M$ at the same height as $A$. The segments $MC$ (horizontal from $M$ to $C$) and $CA$ (vertical from $C$ down to $A$) are drawn, completing a rectangle $MCFA$. This rectangle is the key visual device: it shows that $AF = AC$ (by the definition of a parabola — every point on the parabola is equidistant from the focus and the directrix). Since $AC$ is the horizontal distance from $C$ to $A$, and $C$ lies on the directrix line $x = -a$, we have $AC = a - (-a) = 2a$. Therefore $AF = 2a$. By symmetry across the x-axis, $AF = FB$, so the full length of the latus rectum $AB = AF + FB = 2a + 2a = 4a$.

Here $a$ is the distance from the vertex to the focus (and also from the vertex to the directrix). For a parabola $y^2 = 4ax$, the latus rectum is always $4a$ — a fixed number that depends only on $a$, not on which point on the parabola you pick. This result is used directly in problems: for example, in Example 5 of the textbook, $y^2 = 8x$ gives $a = 2$, so the latus rectum length is $4 \times 2 = 8$. …

Fig. 10.19 is a clean, standard plot of the parabola $y^2 = 8x$ drawn on the usual $x$–$y$ coordinate axes. The curve opens to the right, with its vertex at the origin $O(0,0)$. The $x$-axis is the axis of symmetry of the parabola — the curve is symmetric about it. The focus is marked at $(2,0)$, and the directrix is drawn as a vertical blue line at $x = -2$. The latus rectum is the chord through the focus perpendicular to the axis; its endpoints lie on the parabola, and its total length is $8$ units (from $y = -4$ to $y = 4$ at $x = 2$).

The physical idea the figure teaches is the focus-directrix definition of a parabola: every point on the curve is equidistant from the focus $F$ and the directrix $d$. For $y^2 = 8x$, the distance from any point $P(x,y)$ on the parabola to the focus $(2,0)$ equals its perpendicular distance to the line $x = -2$. The vertex $O$ is the midpoint of the perpendicular from the focus to the directrix, so it lies exactly halfway between them — at $x = 0$.

The key formula the textbook develops with this figure is the latus rectum length:

For $y^2 = 8x$, comparing with $y^2 = 4ax$ gives $4a = 8$, so $a = 2$. The derivation in the text shows why: by the definition of the parabola, $AF = AC$ where $A$ is the vertex, $F$ the focus, and $C$ the foot of the perpendicular from $A$ to the directrix. Since $AC = FM = 2a$ (the distance from the focus to the directrix is $2a$), we get $AF = 2a$. By symmetry, the full chord $AB$ is twice that, hence $4a$. …