Calculator Inputs
Choose a polar conic setup. The tool converts it into a rectangular equation with clear coefficients and classification.
Formula Used
The calculator starts with the polar conic form below.
r = K / (1 ± e cos θ) or r = K / (1 ± e sin θ)
Here, K may equal e multiplied by d. The value e is eccentricity. The value d is the directrix distance. Use x = r cos θ, y = r sin θ, and r2 = x2 + y2.
For the cosine form, the rectangular equation is:
(1 − e2)x2 + y2 ± 2eKx − K2 = 0
For the sine form, the rectangular equation is:
x2 + (1 − e2)y2 ± 2eKy − K2 = 0
How to Use This Calculator
- Select the trig axis used in your polar conic.
- Choose the plus or minus sign from the denominator.
- Enter eccentricity. This controls the conic type.
- Choose e × d or custom K for the numerator.
- Set decimal precision for cleaner displayed coefficients.
- Press the convert button to view the rectangular equation.
Example Data Table
| Polar Form | e | K | Type | Rectangular Form |
|---|---|---|---|---|
| r = 6 / (1 + 0.75 cos θ) | 0.75 | 6 | Ellipse | 0.4375x² + y² + 9x − 36 = 0 |
| r = 4 / (1 − 1 sin θ) | 1 | 4 | Parabola | x² − 8y − 16 = 0 |
| r = 5 / (1 + 1.2 cos θ) | 1.2 | 5 | Hyperbola | −0.44x² + y² + 12x − 25 = 0 |
Understanding Conic Rectangular Conversion
A conic in polar form often uses a focus at the pole. That pole becomes the origin in the rectangular plane. The equation describes points by distance and angle. Rectangular form describes the same points with x and y coordinates. This calculator links both views through standard substitutions.
The most common polar conic form is r equals K divided by one plus or minus e times cosine or sine. The cosine version places the directrix parallel to the y-axis. The sine version places the directrix parallel to the x-axis. The sign controls which side of the focus the directrix uses.
Eccentricity is the main classifier. When e is less than one, the conic is an ellipse. When e equals one, the conic is a parabola. When e is greater than one, the conic is a hyperbola. A zero value creates a circle when K stays positive.
The conversion starts by multiplying both sides by the denominator. Then r cosine theta becomes x. Likewise, r sine theta becomes y. The remaining r term becomes the square root of x squared plus y squared. Squaring both sides removes the radical. After expansion, like terms are collected.
Cosine equations change the x squared coefficient. Sine equations change the y squared coefficient. The linear term also follows the chosen axis. This helps you check the result quickly. If your original equation uses cosine, expect an x term. If it uses sine, expect a y term.
Completed form gives more geometric detail. For ellipses, it helps show the center and axis lengths. For hyperbolas, it helps locate the center and opening direction. For parabolas, it gives the vertex form. These details are useful for graphing and classroom work.
The numerator can be entered in two ways. Use e times d when your problem gives a directrix distance. Use custom K when the equation already shows a numerator. Both methods lead to the same rectangular process. The calculator also reports the implied directrix when possible.
Decimal precision is useful because conic coefficients can become long. More decimals give greater detail. Fewer decimals make the equation easier to read. Always keep enough precision for graded work. Exact fractional work may still be required in formal proofs.
Check the source equation before entering values. A missed sign changes the graph. A wrong axis changes the linear term. Keep units consistent for distance values. When d is given, use the e times d option. When the numerator is already shown, use custom K. This avoids duplicate multiplication. It also keeps the directrix report meaningful for review. Compare the final equation with your textbook example afterward for accuracy.
This tool is designed for checking algebra, preparing graphs, and comparing forms. It does not replace understanding. Read the steps after each conversion. They show why the rectangular equation matches the polar conic. Careful setup makes each conversion accurate, clear, and reliable.
FAQs
1. What does this calculator convert?
It converts polar conic equations into rectangular equations. The output uses x and y. It also shows classification, directrix details, and helpful algebra steps.
2. Which polar conic form is supported?
It supports r = K divided by 1 plus or minus e cos θ. It also supports the matching sine form. You can enter K directly or calculate it from e and d.
3. What is eccentricity?
Eccentricity measures how open or stretched a conic is. Values below one make ellipses. A value of one makes a parabola. Values above one make hyperbolas.
4. What is K in the formula?
K is the numerator in the polar equation. In many textbook forms, K equals e multiplied by d. Here d is the directrix distance.
5. When should I choose cosine?
Choose cosine when your denominator contains cos θ. This usually gives a horizontal orientation. The rectangular equation will contain a linear x term.
6. When should I choose sine?
Choose sine when your denominator contains sin θ. This usually gives a vertical orientation. The rectangular equation will contain a linear y term.
7. Why does the calculator square both sides?
The conversion creates a square root from r. Squaring removes that radical. Then the equation can be expanded and collected in rectangular form.
8. Can this calculator classify the conic?
Yes. It classifies the conic from eccentricity. It labels circle, ellipse, parabola, or hyperbola based on the entered value.
9. Why is completed form useful?
Completed form reveals graph features. It can show a center, vertex, or opening direction. This makes the converted equation easier to graph.
10. Does the sign change the result?
Yes. The sign changes the linear term. It also changes the side where the implied directrix sits. Select the sign exactly as shown in your equation.
11. Can I print the result?
Yes. Use the download button to open the print dialog. You can save the page as a PDF from most modern browsers.