Finding Points That Define The Latus Rectum Calculator

Calculate Latus Rectum Endpoints

Enter the vertex (h, k) and the value of 4p from the standard equation of the parabola to find the endpoints of the latus rectum.

What is the Latus Rectum and Finding its Endpoints?

The latus rectum of a parabola is a line segment passing through the focus of the parabola, perpendicular to the axis of symmetry, with both endpoints lying on the parabola. The term "latus rectum" is Latin, meaning "side" and "straight," respectively. The length of the latus rectum is always equal to `|4p|`, where `p` is the distance from the vertex to the focus and from the vertex to the directrix.

Finding the points that define the latus rectum (i.e., its endpoints) is crucial for understanding the "width" of the parabola at its focus and for accurately graphing it. Our latus rectum calculator helps you find these endpoints quickly.

This latus rectum calculator is useful for students studying conic sections, engineers, and anyone working with parabolic shapes.

Common Misconceptions

• The latus rectum is NOT parallel to the directrix, but it is perpendicular to the axis of symmetry.
• The length of the latus rectum is `|4p|`, not `4p` (it's always positive).
• The focus is the midpoint of the latus rectum.

Latus Rectum Formula and Mathematical Explanation

A parabola is defined as the set of all points equidistant from a point (the focus) and a line (the directrix). The standard equations for a parabola with vertex (h, k) are:

1. (y - k)² = 4p(x - h): Parabola opens horizontally (right if p > 0, left if p < 0).
   • Vertex: `(h, k)`
   • Focus: `(h + p, k)`
   • Axis of Symmetry: `y = k`
   • Directrix: `x = h - p`
   • Latus Rectum passes through `(h+p, k)` and is vertical. Its endpoints are found by setting `x = h + p` in the equation: `(y - k)² = 4p²`, so `y - k = ±2p`, giving endpoints `(h + p, k + 2p)` and `(h + p, k - 2p)`.
2. (x - h)² = 4p(y - k): Parabola opens vertically (up if p > 0, down if p < 0).
   • Vertex: `(h, k)`
   • Focus: `(h, k + p)`
   • Axis of Symmetry: `x = h`
   • Directrix: `y = k - p`
   • Latus Rectum passes through `(h, k+p)` and is horizontal. Its endpoints are found by setting `y = k + p` in the equation: `(x - h)² = 4p²`, so `x - h = ±2p`, giving endpoints `(h + 2p, k + p)` and `(h - 2p, k + p)`.

The length of the latus rectum is `|4p|` in both cases.

Variables Table

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the vertex	Units of length	Any real number
k	y-coordinate of the vertex	Units of length	Any real number
p	Distance from vertex to focus/directrix	Units of length	Any non-zero real number
4p	Coefficient in the standard equation, related to latus rectum length	Units of length	Any non-zero real number

Variables used in parabola equations.

Practical Examples (Real-World Use Cases)

Example 1: Parabola Opening Right

Suppose we have the parabola defined by the equation `(y - 2)² = 8(x - 1)`.

Here, `h = 1`, `k = 2`, and `4p = 8`, so `p = 2`. The parabola opens right (since `4p > 0`).

• Vertex: `(1, 2)`
• Focus: `(h + p, k) = (1 + 2, 2) = (3, 2)`
• Latus Rectum Length: `|8| = 8`
• Endpoints of Latus Rectum: `(h + p, k ± 2p) = (3, 2 ± 4)`, so `(3, 6)` and `(3, -2)`.

Our latus rectum calculator would confirm these points.

Example 2: Parabola Opening Down

Consider the parabola `(x + 1)² = -12(y - 3)`.

Here, `h = -1`, `k = 3`, and `4p = -12`, so `p = -3`. The parabola opens down (since `4p < 0`).

• Vertex: `(-1, 3)`
• Focus: `(h, k + p) = (-1, 3 - 3) = (-1, 0)`
• Latus Rectum Length: `|-12| = 12`
• Endpoints of Latus Rectum: `(h ± 2p, k + p) = (-1 ± (-6), 0)`, so `(-7, 0)` and `(5, 0)`.

Using the latus rectum calculator with these inputs gives the correct endpoints.

How to Use This Latus Rectum Calculator

1. Select Equation Type: Choose the standard form of your parabola's equation from the dropdown menu. This determines whether it opens horizontally or vertically.
2. Enter Vertex Coordinates (h, k): Input the values of `h` and `k` from your equation.
3. Enter the Value of 4p: Input the coefficient `4p` directly from your equation. It can be positive or negative.
4. View Results: The calculator instantly displays the endpoints of the latus rectum, the vertex, focus, `p` value, and the length of the latus rectum.
5. See the Graph: A simple graph shows the relative positions of the vertex, focus, and latus rectum endpoints.
6. Reset or Copy: Use the "Reset" button to clear inputs or "Copy Results" to copy the output.

The latus rectum calculator provides immediate feedback, helping you understand the geometry of the parabola.

Key Factors That Affect Latus Rectum Results

The position and orientation of the latus rectum and its endpoints are determined by:

• Value of 'h': Shifts the parabola and its latus rectum horizontally.
• Value of 'k': Shifts the parabola and its latus rectum vertically.
• Value of '4p' (and thus 'p'):
  • Magnitude of 'p': Determines the "width" of the parabola at the focus (length of latus rectum = |4p|). A larger |p| means a wider parabola and a longer latus rectum.
  • Sign of 'p': Determines the direction the parabola opens (and thus the orientation of the latus rectum relative to the vertex along the axis of symmetry).
• Orientation of the Parabola: Whether it opens up/down or left/right, determined by which term is squared in the equation, affects whether the latus rectum is horizontal or vertical.
• Vertex Position: The vertex `(h,k)` is the starting point for locating the focus and latus rectum.
• Focus Position: The latus rectum passes through the focus, so the focus coordinates directly influence the latus rectum's position.

Understanding these factors is key to interpreting the output of the latus rectum calculator.

Frequently Asked Questions (FAQ)

What is the latus rectum?

The latus rectum of a parabola is the chord through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its length is |4p|.

How do you find the endpoints of the latus rectum?

For (y-k)²=4p(x-h), endpoints are (h+p, k±2p). For (x-h)²=4p(y-k), endpoints are (h±2p, k+p). Our latus rectum calculator does this for you.

Can 'p' be negative?

Yes, if 'p' is negative, the parabola opens to the left (for y-squared form) or downwards (for x-squared form).

Can '4p' be zero?

No, if 4p=0, the equation degenerates and does not represent a parabola.

What is the relationship between the focus and the latus rectum?

The focus is the midpoint of the latus rectum.

Why is the length of the latus rectum |4p|?

It comes from substituting the coordinate of the focus into the parabola's equation and solving for the distance between the two points on the parabola that share that coordinate (perpendicular to the axis).

Does every parabola have a latus rectum?

Yes, every parabola has a latus rectum defined by its focus and the parameter 'p'.

How does the latus rectum calculator handle different equation forms?

The calculator asks you to select the form of the equation first, then uses the appropriate formulas based on your selection and the values of h, k, and 4p.

Related Tools and Internal Resources

• Parabola Equation Calculator: Find the equation of a parabola given certain properties.
• Vertex Form Calculator: Convert quadratic equations to vertex form.
• Focus and Directrix Calculator: Find the focus and directrix from the parabola's equation.
• Conic Sections Identifier: Determine if an equation represents a circle, ellipse, parabola, or hyperbola.
• Distance Formula Calculator: Calculate the distance between two points, useful for verifying latus rectum length.
• Midpoint Calculator: Find the midpoint of a line segment.