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 (or go straight ahead, we won't stop you), 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.

Figures described by the intersection of a plane with a cone (circle, ellipse, parabola, hyperbola).
Figures described by the intersection of a plane with a cone's surface at different angles.

As you can see in the figure above, three shapes can be formed:

But we will only be interested in the parabola, hyperbola, and ellipse. Why? Because they have at least one focus and directrix.

🔎 Every point in a conic section is equidistant to a point (focus) and a directrix (line), both of which can be located anywhere in space.

How does that relate to the latus rectum? Let's see how below.

What is the latus rectum?

The latus rectum is the line segment that goes through a focus and is parallel to the directrix of a conic section. It comes from the Latin words 'latus', which means 'side', and 'rectum', which means 'straight'.

Parabola graph with the latus rectum crossing through the focus while being parallel to the directrix.
Latus rectum for a parabola.

The latus rectum endpoints are on the curve, and you can calculate its length and endpoints' coordinates with our latus rectum calculator.

Latus rectum formula

The equation for the latus rectum depends on the shape you wish to calculate it for, and, as such, we will have:

  • Latus rectum of a parabola;
  • Latus rectum of a hyperbola; and
  • Latus rectum of an ellipse.

Let's take a look at the latus rectum formula for each curve.

Latus rectum of a parabola

Let aa be the distance between the focus and the vertex of a parabola, the latus rectum (lrlr) formula for a parabola is:

lr=4a\quad lr=4a

Latus rectum of a hyperbola

The standard equation for a hyperbola is:

(xh)2a2(yk)2b2=1\quad \frac{(x-h)^2}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1

And using the same parameters aa and bb, we obtain the latus rectum of a hyperbola:

lr=2b2a\quad lr = 2\frac{b^{2}}{a}

Latus rectum of an ellipse

Similarly, the equation for an ellipse is:

(xh)2a2+(yk)2b2=1\quad \frac{(x-h)^2}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1

And the latus rectum formula is the same as the hyperbola:

lr=2b2a\quad lr = 2\frac{b^{2}}{a}

How to find the latus rectum endpoints

With our latus rectum calculator, you can easily find the endpoints for the latus rectum, or you can go the hard way and solve the equations manually 😅 (it's your choice).

For a parabola with vertex at (h,k)(h,k) its endpoints will be located at:

  • (h+lr2,k±lr)(h+\frac{lr}{2},k\pm lr), if it opens to the right;
  • (hlr2,k±lr)(h-\frac{lr}{2},k\pm lr), if it opens to the left;
  • (h±lr,k+lr2)(h\pm lr,k + \frac{lr}{2}), if it opens upward; or
  • (h±lr,klr2)(h\pm lr,k - \frac{lr}{2}), if it opens downward.

For a horizontal hyperbola:

  • (h+c,k±lr2)(h + c,k \pm \frac{lr}{2}); and
  • (hc,k±lr2)(h - c,k \pm \frac{lr}{2}) .

For a vertical hyperbola:

  • (h±lr2,k+c)(h \pm \frac{lr}{2},k + c); and
  • (h±lr2,kc)(h \pm \frac{lr}{2},k - c) .

Where cc is the linear eccentricity of a hyperbola c=a2+b2c = \sqrt{a^{2}+b^{2}}.

And finally, for a horizontal ellipse:

  • (hc,k±lr2)(h-c,k\pm \frac{lr}{2}); and
  • (h+c,k±lr2)(h+c,k\pm \frac{lr}{2}).

And for a vertical ellipse:

  • (h±lr2,k+c)(h\pm \frac{lr}{2},k+c); and
  • (h±lr2,kc)(h\pm \frac{lr}{2},k-c).

Where c=a2b2c = \sqrt{a^{2}-b^{2}}.

Solving problems with the calculator

Let's see how you can use our latus rectum calculator to find the latus rectum for each shape.

Parabola example

Suppose we have a parabola described by the following equation:

y=4x22x+6\quad y=4x^{2}-2x+6

This is a vertical parabola. Using our latus rectum calculator, we just need to identify each parameter:

  1. A=4A = 4.
  2. B=2B = -2.
  3. C=6C = 6.
  4. Plug in the data and the calculator will solve it for you! lr=0.25lr = 0.25.

Or, alternatively by hand:

  1. This parabola has a vertex at (14,234)(\frac{1}{4}, \frac{23}{4}), and its focus is located at (14,9316)(\frac{1}{4},\frac{93}{16}).

  2. And so, the distance between the two points is 0.06250.0625.

  3. Using the latus rectum of a parabola formula, we multiply the result by 4 to obtain: lr=0.25lr = 0.25.

Hyperbola example

We have a hyperbola with the following equation:

(x2)29y24=1\quad\frac{(x-2)^{2}}{9}-\frac{y^{2}}{4} = 1

This is equal to:

(x2)232(y0)222=1\quad\frac{(x-2)^{2}}{3^{2}}-\frac{(y-0)^{2}}{2^{2}} = 1

Since the term containing the xx is positive, this is a horizontal hyperbola.

We identify the parameters:

  1. Transverse axis (a)=3(a) = 3.
  2. Conjugate axis (b)=2(b) = 2.
  3. h=2h = 2.
  4. k=0k = 0.
  5. We plug in and obtain: lr=2.6666...lr = 2.6666....

Ellipse example

Assume we have an ellipse described by:

x225+y27=1\quad\frac{x^{2}}{25}+\frac{y^{2}}{7}=1

The parameters are:

  1. a=5a=5.
  2. b=72.6458b=\sqrt{7} \sim 2.6458
  3. h=0h=0.
  4. k=0k=0.
  5. Resulting in: lr=2.8001lr= 2.8001.

FAQ

What is the latus rectum's length?

  • 4 × a for a parabola, where a is the distance between the focus and the vertex.
  • 2 × b²/a for a hyperbola or ellipse, where a and b are the major and minor axis, respectively.

How do I find latus rectum's endpoints for a vertical parabola?

To find the latus rectum endpoints for a vertical parabola:

  1. Write down the vertex coordinates (h, k) and latus rectum's length lr.
  2. Check if the leading coefficient a is positive or negative.
  3. Replace the vertex coordinates in (h ± lr, k + lr/2) if a is positive, or in (h ± lr, k - lr/2) if it's negative.

Do all conic sections have a latus rectum?

Yes, all conic sections have a latus rectum. While some will have two latus rectum (hyperbola and ellipse), others will have just one (parabola and circle).

Luciano Mino
Shape
Parabola
Orientation
Vertical
Equation: y = Ax² + Bx + C
A
B
C
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