Calculator Inputs
Example Data Table
| Model | Input | Formula Used | Cartesian Result |
|---|---|---|---|
| Line | y = 2x + 3 | -mx + y - b = 0 | -2x + y - 3 = 0 |
| Circle | Center (1, -2), r = 5 | x² + y² - 2hx - 2ky + h² + k² - r² = 0 | x² + y² - 2x + 4y - 20 = 0 |
| Ellipse | h = 0, k = 0, a = 4, b = 2 | (x - h)² / a² + (y - k)² / b² = 1 | 4x² + 16y² - 64 = 0 |
| Parabola | h = 1, k = 0, p = 2 | (x - h)² = 4p(y - k) | x² - 2x - 8y + 1 = 0 |
Formula Used
Rectangular coordinates and Cartesian coordinates use the same coordinate plane. The calculator therefore rewrites the selected rectangular model into a recognized Cartesian equation.
- Line: y = mx + b becomes -mx + y - b = 0.
- Circle: (x - h)² + (y - k)² = r².
- Ellipse: (x - h)² / a² + (y - k)² / b² = 1.
- Horizontal hyperbola: (x - h)² / a² - (y - k)² / b² = 1.
- Vertical hyperbola: (y - k)² / a² - (x - h)² / b² = 1.
- Vertical parabola: (x - h)² = 4p(y - k).
- Horizontal parabola: (y - k)² = 4p(x - h).
How to Use This Calculator
- Select the equation model from the dropdown list.
- Enter the known values for that model.
- Press the calculate button.
- Read the Cartesian equation below the header.
- Review the formula and steps.
- Use CSV or PDF export for saving your result.
Rectangular and Cartesian Equations Explained
A rectangular equation uses x and y coordinates. A Cartesian equation also uses x and y coordinates. For many classroom tasks, the two names describe the same coordinate language. The useful step is often not a true conversion. It is a clean rewrite into a standard Cartesian form.
Why Standard Form Helps
Standard form makes a graph easier to read. A line can become Ax plus By plus C equals zero. A circle can show its center and radius. A parabola can show its vertex and opening direction. A conic can reveal its shape before graphing. This calculator organizes those changes with clear steps.
Practical Uses
Students can check algebra homework. Teachers can prepare examples. Designers can model arcs or paths. Analysts can compare equations in a shared format. The tool also helps when a problem gives a rectangular expression, but asks for a Cartesian equation.
How the Calculator Works
First, choose the equation model. Then enter the known values. The tool applies the matching identity. It expands squares when needed. It also keeps a standard form when that form is more readable. Each result includes the main equation, an alternate expression, and a short explanation.
Lines and Conics
For a line, the calculator rewrites slope or point data into general Cartesian form. For a circle, it expands the center-radius equation. For an ellipse or hyperbola, it uses squared distances from the center. For a parabola, it uses the vertex form with parameter p.
Accuracy Tips
Use consistent units. Check signs carefully. Negative centers change expansion signs. A zero radius is not a circle. Equal points cannot define a unique line. For conics, a and b should be positive. For parabolas, p controls direction and width.
Export Benefits
The CSV download is useful for spreadsheets. The PDF download is helpful for sharing work. Both exports keep the equation, steps, and important values together. This makes review faster and reduces copying errors.
Use the examples to compare inputs, outputs, and formulas before entering your own values.
Final Note
Because rectangular coordinates are already Cartesian, the best result is a simplified, recognized, and well labeled Cartesian equation.
FAQs
1. Are rectangular and Cartesian equations the same?
Yes. Rectangular equations usually use x and y on the Cartesian coordinate plane. This calculator mainly standardizes and expands them.
2. What does this calculator convert?
It converts common rectangular models into standard Cartesian equations. It handles lines, circles, ellipses, hyperbolas, parabolas, and direct equations.
3. Can I use it for conic sections?
Yes. It supports circles, ellipses, hyperbolas, and parabolas. It also shows the standard formula used for each model.
4. Why is my result in general form?
General form places all terms on one side. It is useful for comparison, graphing, classification, and algebra checks.
5. What values should a and b have?
For ellipses and hyperbolas, a and b should be positive. They represent semi-axis values in the selected standard equation.
6. What does p mean in a parabola?
The value p controls the parabola width and direction. Positive and negative values create openings in opposite directions.
7. Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a readable report.
8. Does this replace graphing?
No. It prepares the equation for graphing. You can use the result with graphing software or manual sketching.