Latus Rectum Calculator — Find the Latus Rectum of a Parabola, Ellipse or Hyperbola and Its Endpoints

Latus Rectum Calculator

Our latus rectum calculator will obtain the latus rectum of a parabola, hyperbola, or ellipse and their respective endpoints from just a few parameters describing your function.

If you're wondering what the latus rectum is or how to find the latus rectum, you've come to the right place. We will cover those questions (and more) below, paired with some examples for each conic section. Keep reading to learn more!

Conic sections

Before we jump on the equation of the latus rectum, let's take a look at the concept from where it is derived: a conic section.

Conic sections are, in simple terms, the figure described by the intersection of a cone's surface and a plane. Depending on the angle at which the plane slices the cone, four shapes can be formed:

• A circle;
• A parabola;
• A hyperbola; or
• An ellipse.

We will only be interested in the parabola, hyperbola, and ellipse. Why? Because they have at least one focus and directrix. (The circle is really just an ellipse whose two foci have collapsed onto the center — the calculator handles it as a special case.)

🔎 Every point in a conic section is equidistant to a point (the focus) and a directrix (a line), both of which can be located anywhere in space. The ratio of those two distances is the eccentricity, e, and it is what tells the conics apart:

What is the latus rectum?

The latus rectum (from Latin, "straight side") is the chord of a conic section that passes through a focus and is perpendicular to the major (or transverse) axis, with both of its ends lying on the curve.

In other words: stand at the focus, walk perpendicular to the axis in both directions until you hit the curve, and the segment you traced is the latus rectum. Its length, L, measures how "wide" the conic is right at the focus — it is the natural width scale of the curve.

Half of it is called the semi-latus rectum, written ℓ = L / 2. The semi-latus rectum is arguably the more fundamental quantity, because every conic can be written in one single polar equation:

r = ℓ / (1 + e · cos θ)

where ℓ is the semi-latus rectum and e is the eccentricity. Feed it e = 0 and you get a circle; 0 < e < 1 an ellipse; e = 1 a parabola; e > 1 a hyperbola. One formula, all the conics.

Latus rectum of a parabola

For a parabola written in vertex form with a horizontal axis:

(y − k)² = 4p(x − h)

the vertex sits at (h, k) and p is the focal distance — the distance from the vertex to the focus. The latus rectum is simply:

L = |4p|

That is a remarkably clean result: the latus rectum of a parabola is four times its focal distance. It also equals the distance from the focus to the directrix, doubled.

• Focus: (h + p, k)
• Directrix: x = h − p
• Endpoints: (h + p, k + 2p) and (h + p, k − 2p)

If p > 0 the parabola opens to the right; if p < 0 it opens to the left. For a vertical axis, use (x − h)² = 4p(y − k) — the roles of x and y simply swap, and the parabola opens up (p > 0) or down (p < 0).

Latus rectum of an ellipse

For an ellipse centered at (h, k) with a horizontal major axis:

(x − h)²/a² + (y − k)²/b² = 1

where a is the semi-major axis and b the semi-minor axis (so a ≥ b), the latus rectum is:

L = 2b² / a

An ellipse has two foci, and therefore two latus recta — one through each focus, both of the same length. The distance from the center to each focus is c = √(a² − b²), and the eccentricity is e = c / a.

• Foci: (h ± c, k)
• Endpoints (at the focus (h + c, k)): (h + c, k + b²/a) and (h + c, k − b²/a)

Notice the special case a = b: then c = 0, e = 0, both foci land on the center, and L = 2a — the diameter of a circle. The formula degrades gracefully.

Latus rectum of a hyperbola

For a hyperbola centered at (h, k) with a horizontal transverse axis:

(x − h)²/a² − (y − k)²/b² = 1

where a is the semi-transverse axis and b the semi-conjugate axis, the latus rectum is — perhaps surprisingly — exactly the same expression as for the ellipse:

L = 2b² / a

The only thing that changes is the focal distance: for a hyperbola, c = √(a² + b²) (a plus sign instead of a minus), which pushes the foci outside the curve and makes e = c/a always greater than 1.

• Foci: (h ± c, k)
• Asymptotes: y − k = ±(b/a)(x − h)
• Endpoints (at the focus (h + c, k)): (h + c, k + b²/a) and (h + c, k − b²/a)

Like the ellipse, a hyperbola has two foci and two latus recta, one on each branch.

How to find the latus rectum endpoints

The recipe is the same for all three conics, and it follows straight from the definition:

1. Locate the focus. For a parabola that's (h + p, k); for an ellipse or a hyperbola it's (h ± c, k).
2. Compute the semi-latus rectum ℓ = L / 2. This is |2p| for a parabola, and b²/a for an ellipse or hyperbola.
3. Step ℓ away from the focus in both directions, perpendicular to the axis. If the axis is horizontal, that means moving in ±y; if it's vertical, in ±x.

So for a horizontal axis the endpoints are (x_focus, y_focus ± ℓ), and for a vertical axis they are (x_focus ± ℓ, y_focus). That's it — no calculus required.

Solving problems with the calculator

Using the latus rectum calculator takes three steps:

1. Pick your conic section — parabola, ellipse, or hyperbola.
2. Choose the axis orientation (horizontal or vertical) and enter the parameters: p for a parabola, or the semi-axes a and b for an ellipse/hyperbola, plus the vertex/center coordinates (h, k).
3. Read off the results — the calculator returns the latus rectum L, the semi-latus rectum ℓ, the eccentricity, the foci, the directrix (or directrices), the asymptotes for a hyperbola, both latus rectum endpoints, and a diagram of the whole thing.

Worked example: parabola

Take (y − 0)² = 8(x − 0), i.e. p = 2:

• L = |4p| = |4 × 2| = 8
• ℓ = 2p = 4
• Focus: (2, 0)  •  Directrix: x = −2
• Endpoints: (2, 4) and (2, −4)

Worked example: ellipse

Take x²/25 + y²/9 = 1, i.e. a = 5, b = 3:

• c = √(25 − 9) = √16 = 4  •  e = 4/5 = 0.8
• L = 2b²/a = 2 × 9 / 5 = 3.6  •  ℓ = 1.8
• Foci: (4, 0) and (−4, 0)
• Endpoints at (4, 0): (4, 1.8) and (4, −1.8)

Worked example: hyperbola

Take x²/25 − y²/9 = 1, i.e. a = 5, b = 3:

• c = √(25 + 9) = √34 ≈ 5.831  •  e ≈ 1.166
• L = 2b²/a = 2 × 9 / 5 = 3.6  •  ℓ = 1.8
• Foci: (±5.831, 0)  •  Asymptotes: y = ±0.6x
• Endpoints at (5.831, 0): (5.831, 1.8) and (5.831, −1.8)

Measurement units and costs

The calculator supports both metric (mm, cm, m, km) and US/Imperial (in, ft, yd, mi) length units — or no unit at all, if you're working with a purely abstract conic.

Because the latus rectum is a real, physical chord, it often has a price tag attached: think of the support strut across a parabolic arch, the feed boom of a satellite dish, or the cross-brace of an elliptical window. Enter an optional cost per unit length and the calculator will price the chord for you in any of 20 world currencies — US dollar, Russian ruble, euro, pound, yen, yuan, and more. For ellipses and hyperbolas, which carry two latus recta, it prices both.

FAQs

What is the latus rectum of a parabola?

It is the chord through the focus perpendicular to the axis of symmetry, and its length equals |4p|, where p is the distance from the vertex to the focus. For y² = 8x, the latus rectum is 8.

Why do the ellipse and the hyperbola share the same latus rectum formula?

Because both are defined by the same underlying relation ℓ = b²/a; the only difference is how c is built — √(a² − b²) for the ellipse versus √(a² + b²) for the hyperbola. That sign flip moves the foci but leaves the width-at-the-focus expression untouched.

Can the latus rectum be negative?

No. The latus rectum is a length, so it is always positive. The parameter p can be negative (it encodes the direction the parabola opens), which is why the formula uses the absolute value: L = |4p|.

What is the latus rectum of a circle?

A circle is an ellipse with a = b, so L = 2b²/a = 2a — the diameter. Both foci sit at the center, so the two latus recta coincide.

How do I find the latus rectum endpoints?

Find the focus, halve the latus rectum to get ℓ, then step ℓ away from the focus in both directions perpendicular to the axis. For a horizontal axis: (x_focus, y_focus ± ℓ).

Related Calculators

You might also be interested in our parabola calculator, ellipse calculator, and hyperbola calculator, or in the polar coordinates and cylindrical coordinates calculators for working with the polar form r = ℓ/(1 + e·cos θ).