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Latus Rectum Calculator

THE LATUS RECTUM
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Type the three coefficients, then press =.
The squares completed, and the curve drawn
Your parabola, ellipse or hyperbola
Paste a parabola, an ellipse or a hyperbola — y = 2x² − 8x + 3, x²/25 + y²/9 = 1 or x²/9 − y²/16 = 1 all work. You are never asked which one it is, and picking the wrong formula is the mistake this page exists to remove. A circle works too — the formula collapses to its diameter.
You are not asked which one it is.
Tap a box, then use the keys
The latus rectum is the chord through a focus at right angles to the axis. It measures how wide the curve is exactly where the focus sits. A parabola has one and its length is |4p|; an ellipse and a hyperbola have two, one at each focus, and both use 2b²/a. Those two share a formula but reach it from opposite signs, which is where the confusion starts.
The working, step by step
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This latus rectum calculator takes any conic and gives you the length of its latus rectum, its endpoints, and how many of them the curve has — and it works out for itself whether you have a parabola, an ellipse or a hyperbola.

Type y = 2x² − 8x + 3 and you get 1/2. Type x²/25 + y²/9 = 1 and you get 18/5, twice over — one at each focus.

What the Latus Rectum Calculator Does Not Ask You

Every other tool for this job asks the same two questions first: which conic is it, and then what are the parameters — a for a parabola, a and b for an ellipse or a hyperbola.

One of those pages then lists, among its own common mistakes, that using the ellipse formula for a parabola gives the wrong answer, and warns you to keep the semi-major and semi-minor axes straight. Both of those mistakes are made possible by asking. Paste the equation and this page decides.

What a Latus Rectum Is

The chord through a focus, at right angles to the axis, with both ends on the curve. It answers one question in one number: how wide is this curve exactly where the focus sits?

That makes it the fastest way to sketch a conic once you have the focus. Mark the focus, step half the length either way, and you have two points of the curve that no amount of guessing would have given you.

How Many You Get

One for each focus. Nothing more to it:

parabola → one focus → one latus rectum
ellipse → two foci → two, of equal length
hyperbola → two foci → two, one on each branch

The two are always equal because the curve is symmetric about its centre.

The Two Formulas

parabola: |4p|
ellipse and hyperbola: 2b²/a

The parabola one needs no work at all. Get the equation to (x − h)² = 4p(y − k) and the coefficient on the right is the length — the same number that fixes the whole shape.

Why an Ellipse and a Hyperbola Share One

This surprises people, and it is the source of most of the confusion. The formula is identical; what differs is how a and b were found.

On an ellipse, a is the larger denominator. On a hyperbola, a is whichever sits under the positive term — and it can easily be the smaller. Put the wrong one on top and the answer is wrong even though the formula was right, which is exactly the trap the market's own pages warn about.

A Circle Is Not a Special Case

Set a equal to b and 2b²/a becomes 2a — the diameter. The two foci have met at the centre, and the chord through them is simply a diameter.

Nothing had to be added for it. That is usually the sign that a formula has been written down properly.

Expect a Surd, and Keep It

a is a square root far more often than it is a whole number, so the length usually is too. For x²/3 + y²/2 = 1 the answer is 4√3/3.

Rounding matters more here than elsewhere, because the latus rectum is often the number a sketch is drawn from. A wrong width at the focus pulls the whole curve out of shape, and the error shows in the drawing rather than hiding in the arithmetic.

When There Is None

A latus rectum needs a focus to pass through. A pair of crossing lines, a single point, a straight line and an empty equation have none, and this page names each rather than guessing.

One more case is worth knowing: an equation with an xy term is still a conic, just a turned one. It does have a latus rectum — calling it nothing would be a wrong answer — but finding it needs a rotation first.

Check the Ends Against the Curve

Whichever formula you used, the two ends of the chord are points of the conic. Put them back into the equation you started with and it must come out to zero.

That one test works for all three curves, which means it catches a wrong formula as easily as a wrong number — and a wrong formula is the mistake this topic is famous for.

Using the Latus Rectum Calculator

Put the equation in the box, in whatever form you have it, and tap the button. You get the length, which conic it turned out to be, which formula that gave, how many chords there are, where each one starts and ends, and a drawing with the chord marked on the curve through every focus.

Latus Rectum Calculator FAQ

What is the latus rectum of a conic?
The chord that passes through a focus at right angles to the axis. It measures how wide the curve is exactly where the focus sits.
What is the formula for the latus rectum?
For a parabola it is the size of four p. For an ellipse and for a hyperbola it is two b squared over a. The last two share a formula but reach a and b from opposite signs.
How many latus rectums does a conic have?
One for each focus. A parabola has one focus so it has one; an ellipse and a hyperbola have two foci and so two chords, of equal length.
Does a circle have a latus rectum?
Yes, and it is the diameter. With a and b equal the formula two b squared over a collapses to two a, and the two foci have met at the centre.
Why is the latus rectum useful?
It gives the width of the curve at the focus in a single number, which is the quickest way to sketch a conic accurately once you know where the focus is.

Work out the conic first, then the formula follows — and put the endpoints back in to be sure.