Calculator Input
Enter coefficients from Ax² + Cy² + Dx + Ey + F = 0. This tool handles axis aligned hyperbolas.
Formula used
Start with the general axis aligned conic form:
For a hyperbola, A and C must have opposite signs. Complete each square.
Divide by the right side. Put the positive squared term first. Then read a², b², vertices, foci, and asymptotes.
How to use this calculator
- Write your equation as Ax² + Cy² + Dx + Ey + F = 0.
- Enter each coefficient in the matching input field.
- Use zero when a linear x or y term is missing.
- Select decimal places for rounded output.
- Press the convert button to view the standard form.
- Download the CSV or PDF result when needed.
Example Data Table
| General equation | Orientation | Standard form | Center |
|---|---|---|---|
| 9x² - 16y² - 54x - 64y - 127 = 0 | Horizontal | (x - 3)² / 16 - (y + 2)² / 9 = 1 | (3, -2) |
| -4x² + 25y² + 24x - 100y - 36 = 0 | Vertical | (y - 2)² / 4 - (x - 3)² / 25 = 1 | (3, 2) |
| 4x² - y² + 8x + 6y - 9 = 0 | Horizontal | (x + 1)² / 1 - (y - 3)² / 4 = 1 | (-1, 3) |
Clear Hyperbola Conversion Guide
A hyperbola can look difficult in general form. The terms may be spread out. The center is hidden. The opening direction is not always clear. Standard form fixes that problem. It places each squared expression around the center. It also shows the transverse axis, conjugate axis, vertices, foci, and asymptotes.
What The Calculator Does
This calculator converts an axis aligned hyperbola equation into standard form. It accepts A, C, D, E, and F from Ax² + Cy² + Dx + Ey + F = 0. The x² and y² coefficients must have opposite signs. That sign pattern separates a hyperbola from an ellipse. The tool completes the square for x and y. Then it moves the remaining constant to the right side.
Why Completing The Square Helps
Completing the square rewrites x² plus a linear x term. It creates a perfect square around x minus h. The same idea works for y. After this step, the center appears as (h, k). The equation then becomes easier to divide by the right side value. Each denominator becomes a² or b².
Reading The Final Form
A horizontal hyperbola has the x square first. Its form is (x - h)² / a² - (y - k)² / b² = 1. A vertical hyperbola has the y square first. Its form is (y - k)² / a² - (x - h)² / b² = 1. The first positive squared term shows the opening direction.
Important Output Values
The calculator gives a², b², a, b, and c. It also gives vertices, foci, and asymptotes. For a horizontal hyperbola, vertices move left and right from the center. For a vertical hyperbola, vertices move up and down. The value c comes from c² = a² + b². Foci always sit along the transverse axis.
Best Uses
Use this tool before graphing. It helps check homework. It also helps teachers prepare examples. Engineers and designers can use it when conic sections appear in layouts or reflections. The step display is useful because it shows the algebra path, not only the final answer.
Input Tips
Enter zero for missing linear terms. Use negative signs with care. Keep x² and y² coefficients nonzero. Use fractions when needed, such as 3/2. Select more decimal places when the coefficients do not divide evenly. If the result says the equation is degenerate, check the signs and constant carefully.
Common Mistakes To Avoid
Do not divide before completing the square. That can hide the correct center. Do not treat a negative denominator as valid standard form. The final denominators must be positive. Also remember that a is not always under x. The positive term decides which variable starts the standard form. Rounding can change labels slightly. Keep exact values in notes when possible. Use the decimal output for quick checking, and use the steps for written work.
A clear setup prevents sign errors during conversion. It also makes graph checks easier for students later.
FAQs
What equation form does this calculator use?
It uses Ax² + Cy² + Dx + Ey + F = 0. This covers axis aligned hyperbolas without an xy term. Rotated hyperbolas need a separate rotation method.
How do I know it is a hyperbola?
The x² and y² coefficients must have opposite signs. One squared term must be positive after normalization. If both signs match, the equation is not an axis aligned hyperbola.
What is standard form for a horizontal hyperbola?
The horizontal form is (x - h)² / a² - (y - k)² / b² = 1. It opens left and right from the center.
What is standard form for a vertical hyperbola?
The vertical form is (y - k)² / a² - (x - h)² / b² = 1. It opens upward and downward from the center.
What does the center mean?
The center is the midpoint between the two vertices. It is written as (h, k). The calculator finds it by completing the square.
Can I enter fractions?
Yes. Enter values like 3/2 or -7/4. The parser also accepts decimals, integers, and scientific notation. Avoid mixed fraction text.
Why does the calculator show degenerate?
It can happen when the completed square right side becomes zero. The equation may represent intersecting lines or another special case, not a standard hyperbola.
What are asymptotes?
Asymptotes are guide lines that the branches approach. They pass through the center. Their slopes come from a and b in the standard form.
What are foci?
Foci are fixed points along the transverse axis. Their distance from the center is c. For every hyperbola, c² = a² + b².
Can this solve rotated hyperbolas?
No. A rotated hyperbola has an xy term. That case needs angle rotation before standard form conversion. This calculator focuses on axis aligned equations.
Why are my denominators decimal values?
Your coefficients may not divide evenly. Increase decimal places for more detail. Keep the step result when you need a cleaner written explanation.