Find Equation Of Parabola Given Focus And Directrix Calculator

What is a Find Equation of Parabola Given Focus and Directrix Calculator?

A find equation of parabola given focus and directrix calculator is a tool used to determine the standard equation of a parabola when you know the coordinates of its focus (a fixed point) and the equation of its directrix (a fixed line). A parabola is defined as the set of all points that are equidistant from the focus and the directrix. This calculator helps students, mathematicians, and engineers quickly find the parabola's equation without manual derivation. Our find equation of parabola given focus and directrix calculator simplifies this process.

Anyone studying conic sections, analytic geometry, or dealing with parabolic reflectors, antennas, or trajectories might use this calculator. Common misconceptions are that the focus is always above the directrix or that the vertex is always at (0,0); neither is necessarily true. The orientation depends on the relative positions of the focus and directrix, and the find equation of parabola given focus and directrix calculator handles these cases.

Find Equation of Parabola Given Focus and Directrix Calculator: Formula and Mathematical Explanation

A parabola is defined by a focus point (h, v) and a directrix line (y=k or x=k). The vertex of the parabola lies halfway between the focus and the directrix.

Case 1: Directrix is y = k

• Focus: F(h, v)
• Directrix: y = k
• The vertex V is at (h, (v+k)/2).
• The distance 'p' from the vertex to the focus (and vertex to directrix) is |v-k|/2. If v > k, p = (v-k)/2 and the parabola opens upwards. If v < k, p = (k-v)/2 (but using 4p in the formula, we take p=(v-k)/2, so 4p is positive if v>k, negative if v
• The standard equation is: (x – h)² = 4p(y – (v+k)/2)

Case 2: Directrix is x = k

• Focus: F(h, v)
• Directrix: x = k
• The vertex V is at ((h+k)/2, v).
• The distance 'p' is related to |h-k|/2. Let's define p = (h-k)/2.
• The standard equation is: (y – v)² = 4p(x – (h+k)/2)

The find equation of parabola given focus and directrix calculator uses these formulas based on the provided inputs.

Variable	Meaning	Unit	Typical Range
(h, v) or (fx, fy)	Coordinates of the Focus	Units	Real numbers
k or directrixValue	Constant defining the Directrix (y=k or x=k)	Units	Real numbers
(vx, vy)	Coordinates of the Vertex	Units	Calculated
p	Directed distance from vertex to focus	Units	Calculated, can be positive or negative
4p	Latus rectum length (absolute value)	Units	Calculated

The find equation of parabola given focus and directrix calculator accurately implements these steps.

Practical Examples

Example 1:

Focus at (2, 3), Directrix y = 1.
Inputs for the find equation of parabola given focus and directrix calculator: fx=2, fy=3, directrixType='y=', directrixValue=1.
Vertex: (2, (3+1)/2) = (2, 2)
p = (3-1)/2 = 1
4p = 4
Equation: (x – 2)² = 4(y – 2)

Example 2:

Focus at (-1, 4), Directrix x = -3.
Inputs for the find equation of parabola given focus and directrix calculator: fx=-1, fy=4, directrixType='x=', directrixValue=-3.
Vertex: ((-1+(-3))/2, 4) = (-2, 4)
p = (-1 – (-3))/2 = 1
4p = 4
Equation: (y – 4)² = 4(x – (-2)) => (y – 4)² = 4(x + 2)

How to Use This Find Equation of Parabola Given Focus and Directrix Calculator

1. Enter Focus Coordinates: Input the x and y coordinates of the focus point (fx, fy).
2. Select Directrix Type: Choose whether the directrix is a horizontal line (y=) or a vertical line (x=).
3. Enter Directrix Value: Input the constant value 'k' for the directrix equation (y=k or x=k).
4. Calculate: The calculator automatically updates, but you can click "Calculate".
5. Read Results: The primary result is the equation of the parabola. Intermediate results show the vertex, 'p' value, and axis of symmetry.
6. View Graph and Table: The graph visualizes the parabola, focus, directrix, and vertex. The table shows sample points.

The results from the find equation of parabola given focus and directrix calculator give you the standard form equation, which is useful for graphing and analysis. For more on conic sections, see our conic sections overview.

Key Factors That Affect Parabola Equation Results

• Focus Coordinates (fx, fy): These directly determine the position of the focus and influence the vertex and 'p'.
• Directrix Equation (y=k or x=k): The type and value of the directrix determine the parabola's orientation (up/down or left/right) and, along with the focus, its vertex and 'p'.
• Relative Position of Focus and Directrix: The distance between the focus and directrix determines the magnitude of 'p' (and 4p), which affects the parabola's width. The side of the directrix the focus lies on determines the sign of 'p' and the direction of opening.
• Vertex Position: Calculated from the focus and directrix, it's the (h,k) in the standard form (using h,k as vertex coords temporarily).
• Value of 'p': The directed distance from vertex to focus; its magnitude affects the latus rectum length (4|p|), and its sign determines the opening direction.
• Axis of Symmetry: A line passing through the focus and vertex, perpendicular to the directrix. It's either x=vx or y=vy. Our find equation of parabola given focus and directrix calculator finds this too. You might also be interested in a parabola vertex calculator.

Frequently Asked Questions (FAQ)

What is a parabola?

A parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix).

What if the focus is on the directrix?

If the focus is on the directrix, p=0, and the "parabola" degenerates into a line passing through the focus and perpendicular to the directrix. Our find equation of parabola given focus and directrix calculator might show 4p=0 in such cases.

How do I know if the parabola opens up, down, left, or right?

If the directrix is y=k, and p>0 (focus above directrix), it opens up; if p<0 (focus below directrix), it opens down. If the directrix is x=k, and p>0 (focus to the right), it opens right; if p<0 (focus to the left), it opens left. The find equation of parabola given focus and directrix calculator's equation form shows this.

Can 'p' be negative?

Yes, 'p' is a directed distance. In our formulation for y=k, p=(v-k)/2, so it's positive if v>k and negative if v

What is the latus rectum?

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

How is the vertex related to the focus and directrix?

The vertex is the midpoint between the focus and the point on the directrix closest to the focus.

Can I use this find equation of parabola given focus and directrix calculator for rotated parabolas?

No, this calculator is for parabolas with a horizontal or vertical directrix (and thus axis of symmetry), not rotated ones.

What is the standard form equation?

For a vertical axis: (x-h)^2 = 4p(y-k); for a horizontal axis: (y-k)^2 = 4p(x-h), where (h,k) is the vertex. Our calculator gives this form, sometimes using vx, vy for vertex coordinates. We also have a standard form parabola calculator.

Related Tools and Internal Resources

• Parabola Vertex Calculator: Find the vertex given the equation.
• Standard Form Parabola Calculator: Convert between different forms of parabola equations.
• Conic Sections Overview: Learn more about parabolas, ellipses, and hyperbolas.
• Graphing Quadratic Functions: Understand how to graph parabolas from their equations.
• Distance Formula Calculator: Useful for understanding the definition of a parabola.
• Midpoint Formula Calculator: Helpful for finding the vertex.