Free Latus Rectum Calculator

This free Latus Rectum Calculator quickly finds the latus rectum length and endpoints for a parabola, hyperbola, or ellipse. With just a few input values — such as coefficients, center coordinates, or axis lengths — the tool outputs both the latus rectum length and the exact coordinates of its endpoints, making it ideal for students, teachers, and professionals working with conic sections.

What Is the Latus Rectum?

In conic section geometry, the latus rectum is a line segment that passes through a focus and runs parallel to the corresponding directrix. Its two endpoints lie on the curve, and the segment itself is perpendicular to the axis of symmetry. The term originates from Latin: latus (side) and rectum (straight). This segment appears in all conic sections that have at least one focus — namely the parabola, hyperbola, and ellipse. While a parabola has a single latus rectum, both the ellipse and the hyperbola have two (one through each focus). The length and orientation of the latus rectum provide useful information about the shape and size of the conic.

Latus Rectum Length Formulas

The formula for the latus rectum length depends on the type of conic section and the parameters that define it.

• Parabola: $lr = 4a$, where $a$ is the distance from the vertex to the focus. For a parabola given in standard form $y = Ax^{2} + Bx + C$, the length can also be expressed as $lr = \dfrac{1}{|A|}$ because the focal distance is $1/(4|A|)$.

• Hyperbola: $lr = \dfrac{2b^{2}}{a}$, with $a$ representing the semi‑transverse axis and $b$ the semi‑conjugate axis.

• Ellipse: $lr = \dfrac{2b^{2}}{a}$, where $a$ is the semi‑major axis and $b$ is the semi‑minor axis.

In both the hyperbola and ellipse formulas, $a$ corresponds to the longer axis (transverse for hyperbola, major for ellipse), while $b$ is the shorter axis.

Finding Latus Rectum Endpoints

The coordinates of the latus rectum endpoints vary with the conic’s centre (or vertex), its orientation, and the linear eccentricity $c$.

• Parabola (vertex at $(h,k)$):

  • Opens upward: $(h \pm lr,\; k + lr/2)$.
  • Opens downward: $(h \pm lr,\; k - lr/2)$.
  • Opens right: $(h + lr/2,\; k \pm lr)$.
  • Opens left: $(h - lr/2,\; k \pm lr)$.

• Horizontal hyperbola (centre $(h,k)$): endpoints at $(h \pm c,\; k \pm lr/2)$, where $c = \sqrt{a^{2} + b^{2}}$.

• Vertical hyperbola: endpoints at $(h \pm lr/2,\; k \pm c)$.

• Horizontal ellipse (centre $(h,k)$, $a > b$): endpoints at $(h \pm c,\; k \pm lr/2)$, with $c = \sqrt{a^{2} - b^{2}}$.

• Vertical ellipse: endpoints at $(h \pm lr/2,\; k \pm c)$.

Applying these formulas manually involves multiple algebraic steps, but the Latus Rectum Calculator automates the entire process, returning accurate coordinates for any orientation.

Worked Examples

Example 1: Parabola

Consider the equation $y = 4x^{2} - 2x + 6$. Input the coefficients $A = 4$, $B = -2$, $C = 6$ into the calculator. The vertex is computed as $(0.25, 5.75)$, and the latus rectum length is $lr = 0.25$. This matches the simplified formula $lr = 1/|A| = 1/4$.

Example 2: Hyperbola

For the hyperbola $\dfrac{(x-2)^{2}}{9} - \dfrac{y^{2}}{4} = 1$, set $a = 3$, $b = 2$, centre $(2,0)$. The calculator gives $lr = \dfrac{2 \cdot 2^{2}}{3} = \dfrac{8}{3} \approx 2.6667$. The linear eccentricity is $c = \sqrt{9 + 4} = \sqrt{13} \approx 3.606$, so the endpoints lie at $(2 \pm \sqrt{13},\; 0 \pm 1.3333)$.

Example 3: Ellipse

Take the ellipse $\dfrac{x^{2}}{25} + \dfrac{y^{2}}{7} = 1$, with $a = 5$, $b = \sqrt{7} \approx 2.6458$, centre $(0,0)$. The latus rectum length is $lr = \dfrac{2 \cdot 7}{5} = \dfrac{14}{5} = 2.8$. Here $c = \sqrt{25 - 7} = \sqrt{18} \approx 4.2426$, and the endpoints are $(\pm 4.2426,\; \pm 1.4)$.

These examples illustrate how the tool handles different conic types, requiring only a few input parameters to provide both the latus rectum length and endpoint coordinates.

Why Use a Dedicated Latus Rectum Calculator?

Calculating the latus rectum length and endpoints by hand can be error‑prone, especially when dealing with eccentricity, centre shifts, and multiple orientation cases. The Latus Rectum Calculator streamlines all these steps, delivering instant, reliable results for parabolas, hyperbolas, and ellipses. It supports both standard quadratic forms and centre‑based equations, making it a versatile tool for anyone studying conic sections or applying them in practical geometry. Because it is free and accessible online, it serves as a convenient check for manual work or a quick reference during problem solving.