Math
Latus Rectum Calculator
Enter a parabola as y = ax^2 + bx + c or in the conic form (x - h)^2 = 4p(y - k), or an ellipse or hyperbola with its semi-axes a and b and center (h, k). mathcheck computes the focal distance, the latus rectum length (|4p| for a parabola, 2b^2/a for an ellipse or hyperbola), and the exact coordinates of its endpoints, along with the foci and eccentricity where they apply.
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Result
Latus rectum length
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- Endpoints
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- Focus / foci
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- Line(s) containing the latus rectum
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- Other features
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What the latus rectum is
The latus rectum (Latin for 'straight side') is the chord of a conic that passes through a focus and runs perpendicular to the The vertical line that splits a parabola into two mirror-image halves.. It measures how wide the curve is at the focus, which makes it a quick way to sketch the shape accurately: plot the The turning point of a parabola, or a corner point of a shape., the focus, and the two latus rectum endpoints, then draw the curve through them.
| Conic | Focal distance | Latus rectum | Where |
|---|---|---|---|
| Parabola | p from the vertex | |4p| | one chord through the single focus |
| Ellipse | c = sqrt(a^2 - b^2) | 2b^2/a | one chord through each of the two foci |
| Hyperbola | c = sqrt(a^2 + b^2) | 2b^2/a | one chord through each focus, on each branch |
| Circle | 0 | 2a (any diameter) | through the center |
y = x^2 - 4x + 3
Vertex (2, -1), p = 1/4, focus (2, -3/4). Latus rectum length |1/a| = 1, endpoints (2 ± 1/2, -3/4) = (3/2, -3/4) and (5/2, -3/4).
Ellipse x^2/25 + y^2/9 = 1
a = 5, b = 3, c = 4. Length 2(9)/5 = 18/5. Endpoints (±4, ±9/5).
How to use this calculator
- Choose the conic and the form in which you have it.
- Enter the The number multiplying a variable, like the 3 in 3x., or a, b, the center, and the orientation.
- Read the latus rectum length, its endpoints, the focus or foci, and the line the chord lies on.
- Open Show the work to see p or c computed and the endpoint formulas substituted.
How to read the answer
The length is exact as a fraction, and the endpoints are exact fractions or A root expression written with the √ symbol, such as a square root. when c is A number that cannot be written as a fraction of two whole numbers, like π or √2.. For a parabola the endpoints sit on the horizontal (or vertical) line through the focus, 2p on either side; for an ellipse or hyperbola each focus has its own latus rectum, and the endpoints are b^2/a above and below (or left and right of) each focus. A circle is the a = b case where the foci merge.
Common mistakes and edge cases
- Using 2p or p as the parabola's latus rectum; the length is |4p| = |1/a|.
- Confusing a and b for an ellipse: a is always the semi-major axis, so 2b^2/a uses the smaller axis on top.
- Placing the endpoints at the vertex instead of at the focus.
- Subtracting b^2 for a hyperbola: c^2 = a^2 + b^2 there, not a^2 - b^2.
- Ignoring the orientation, which swaps the roles of x and y in the endpoint formulas.
Worked examples
Parabola in standard form
y = x^2 - 4x + 3
Latus rectum length
1
Wide parabola
y = (1/8)x^2
Latus rectum length
8
Parabola opening downward
y = -2x^2 + 8
Latus rectum length
1/2 (0.5)
Conic form, opens right
(y - 1)^2 = 8(x - 2), so p = 2
Latus rectum length
8
Ellipse
x^2/25 + y^2/9 = 1
Latus rectum length
18/5 (3.6)
Ellipse with irrational c
x^2/4 + y^2 = 1
Latus rectum length
1
Hyperbola, vertical
(y - 2)^2/9 - (x - 1)^2/16 = 1
Latus rectum length
32/3 (10.6667)
Circle (a = b)
Latus rectum length
6
Ellipse with a < b
Latus rectum length
—
Not a parabola
Latus rectum length
—
Frequently asked questions
What is the latus rectum of a parabola?+
The chord through the focus parallel to the directrix. Its length is |4p|, where p is the distance from the vertex to the focus; for y = ax^2 + bx + c that is |1/a|.
How do I find the endpoints of the latus rectum?+
Start at the focus and move half the length in each direction perpendicular to the axis: (h ± 2p, k + p) for an upward parabola, and (±c, ±b^2/a) for an ellipse or hyperbola centered at the origin.
What is the latus rectum of an ellipse?+
2b^2/a, where a is the semi-major and b the semi-minor axis. Each focus has its own latus rectum, and both have this length.
Is the hyperbola formula the same as the ellipse formula?+
The length is the same expression, 2b^2/a, but the foci are farther out because c^2 = a^2 + b^2 instead of a^2 - b^2.
Why is the latus rectum useful?+
It gives two exact points on the curve at the focus, so with the vertex you can sketch the conic accurately. It also appears in the polar equation of a conic, r = ℓ/(1 + e cos θ), where ℓ is the semi-latus rectum.
What is the semi-latus rectum?+
Half the latus rectum: 2p for a parabola and b^2/a for an ellipse or hyperbola. It is the distance from the focus to the curve measured perpendicular to the axis.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026