Latus Rectum Calculator

In analytical geometry, the latus rectum is an important line segment associated with conic sections such as parabolas, ellipses, and hyperbolas. It helps in determining the width of a conic section at the focus. Our Latus Rectum Calculator allows you to compute this geometric property instantly and accurately.

Latus Rectum Calculator


The latus rectum is a line segment perpendicular to the axis of symmetry of a conic section and passes through its focus. Its length varies depending on the type of conic:

  • Parabola: Distance across the parabola through the focus, parallel to the directrix.
  • Ellipse: Distance through a focus, perpendicular to the major axis.
  • Hyperbola: Distance through a focus, perpendicular to the transverse axis.

It is often used in physics, astronomy, and engineering calculations involving conic sections.

How to Use the Latus Rectum Calculator

Using the calculator is simple and user-friendly:

  1. Select the Type of Conic: Parabola, Ellipse, or Hyperbola.
  2. Enter Required Parameters:
    • Parabola: Coefficient of x2x^2×2 or y2y^2y2
    • Ellipse: Semi-major axis aaa and semi-minor axis bbb
    • Hyperbola: Semi-major axis aaa and semi-minor axis bbb
  3. Click Calculate: The tool instantly computes the latus rectum.
  4. View Results: Get the exact length of the latus rectum for your conic section.

Formulas for Latus Rectum

1. Parabola
For a parabola y2=4axy^2 = 4axy2=4ax or x2=4ayx^2 = 4ayx2=4ay: Latus Rectum (LR)=4a\text{Latus Rectum (LR)} = 4aLatus Rectum (LR)=4a

2. Ellipse
For an ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1a2x2​+b2y2​=1: Latus Rectum (LR)=2b2a(through each focus)\text{Latus Rectum (LR)} = \frac{2b^2}{a} \quad \text{(through each focus)}Latus Rectum (LR)=a2b2​(through each focus)

3. Hyperbola
For a hyperbola x2a2−y2b2=1\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1a2x2​−b2y2​=1: Latus Rectum (LR)=2b2a(through each focus)\text{Latus Rectum (LR)} = \frac{2b^2}{a} \quad \text{(through each focus)}Latus Rectum (LR)=a2b2​(through each focus)

Where:

  • aaa = Semi-major axis
  • bbb = Semi-minor axis

Example Calculations

Example 1: Parabola

  • Equation: y2=12xy^2 = 12xy2=12x
  • 4a=12  ⟹  a=34a = 12 \implies a = 34a=12⟹a=3
  • Latus Rectum = 4a=124a = 124a=12 units

Example 2: Ellipse

  • Equation: x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 19×2​+4y2​=1
  • a=3,b=2a = 3, b = 2a=3,b=2
  • Latus Rectum = 2b2/a=2×4/3=8/3≈2.672b^2 / a = 2 \times 4 / 3 = 8/3 \approx 2.672b2/a=2×4/3=8/3≈2.67 units

Example 3: Hyperbola

  • Equation: x216−y29=1\frac{x^2}{16} – \frac{y^2}{9} = 116×2​−9y2​=1
  • a=4,b=3a = 4, b = 3a=4,b=3
  • Latus Rectum = 2b2/a=2×9/4=18/4=4.52b^2 / a = 2 \times 9 / 4 = 18 / 4 = 4.52b2/a=2×9/4=18/4=4.5 units

Additional Insights

  1. Focus Relation: The latus rectum passes through the focus, helping determine focal width.
  2. Graphical Applications: Useful for plotting precise conic section curves.
  3. Astronomical Use: Helps in calculating planetary orbits described as ellipses.
  4. Engineering Applications: Used in optical systems, reflectors, and parabolic antennas.
  5. Quick Calculations: Avoids manual derivation with instant calculator results.

20 FAQs About Latus Rectum Calculator

  1. What is a latus rectum?
    A line segment passing through the focus, perpendicular to the axis of symmetry of a conic.
  2. Why is it important?
    It helps measure the width of conic sections at the focus.
  3. Can it be used for parabolas?
    Yes, it calculates 4a4a4a for a parabola.
  4. Can it calculate for ellipses?
    Yes, it uses the formula 2b2/a2b^2 / a2b2/a.
  5. Does it work for hyperbolas?
    Yes, using the same formula 2b2/a2b^2 / a2b2/a.
  6. Is it useful in physics?
    Yes, for optics, motion along conic paths, and planetary orbits.
  7. Does it require the equation of the conic?
    Yes, or the values of aaa and bbb depending on the conic type.
  8. Can it handle decimals?
    Yes, the calculator provides accurate decimal results.
  9. Is it suitable for students?
    Absolutely, especially for geometry, calculus, and analytical geometry studies.
  10. Can it visualize the latus rectum?
    Some calculators provide graphical representation of the conic and latus rectum.
  11. Is it free?
    Yes, online calculators are generally free to use.
  12. Can it handle rotated conics?
    Most basic calculators work with standard orientation; advanced tools may handle rotation.
  13. Does it apply to 3D shapes?
    No, it is specific to 2D conic sections.
  14. Can it be used for orbit calculations?
    Yes, for elliptical orbits and focal distances.
  15. What is the relation between latus rectum and focus?
    The latus rectum always passes through the focus of the conic.
  16. Does it calculate for multiple conics at once?
    Usually, one conic calculation at a time is supported.
  17. Can it help in plotting parabolas?
    Yes, knowing the latus rectum aids in accurate graphing.
  18. Is the formula same for vertical and horizontal parabolas?
    Yes, the length remains 4a4a4a.
  19. How does it help in optics?
    Parabolic reflectors focus light using latus rectum properties.
  20. Can it calculate approximate or exact values?
    It provides both exact and decimal approximations.

Our Latus Rectum Calculator is an essential tool for students, mathematicians, engineers, and astronomers. It simplifies the calculation of latus rectum for parabolas, ellipses, and hyperbolas, enabling accurate geometric analysis and practical applications.