General to Standard Form of an Ellipse Calculator

Enter the general equation (or any equivalent form) of the ellipse to get its standard form and the step-by-step solution.

Quick Examples

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How to Use the Calculator

This online calculator allows you to convert any equation of an ellipse into its standard form, showing the entire algebraic procedure step-by-step. To use this tool, follow these steps:

1. Enter the math equation.

Click on the main input box of the calculator and type the equation of the ellipse you want to analyze. You do not need to sort the equation; the system can interpret the terms regardless of the order they are written in. It is not mandatory for the equation to be in its strict general form (set to zero); you can enter mixed terms on both sides of the equal sign (for example: 36y + 9y^2 + 4x^2 = 24x - 36) and the system will group them correctly.

2. Run the calculation.

Press the "Enter" key on your keyboard or click the main button so the math engine can process the expression.

3. Review the procedure.

The tool will return the equation in its exact standard form. Right below it, the calculator will display the detailed analytical method, explaining how the terms were grouped, how the common factor was factored out, and how the perfect square trinomials were completed to reach the final result.

Note: the calculator seamlessly accepts and processes integer, decimal, and fractional coefficients, as well as irrational numbers (such as exact square roots).

Solved Exercises

The following are examples of problems solved by the calculator.

Convert the general form $ 4x^2+9y^2-24x+36y+36=0 $ to the standard form of ellipse.

Standard form of the ellipse

$$ \dfrac{\left(x - 3\right)^2}{9} + \dfrac{\left(y + 2\right)^2}{4} = 1 $$

For a complete analysis of the ellipse with its elements and graph, check out the ellipse calculator.

Step-by-step solution

1. Identify and prepare the equation.

We start from the entered equation:

$$ 4x^2+9y^2-24x+36y+36=0 $$

Group the terms by variable and move the constant term to the right side of the equation:

$$ \left(4 x^{2}-24 x\right) + \left(9 y^{2}+36 y\right) = -36 $$

2. Factor out the leading coefficients.

Factor out the common factor from the quadratic terms so they have a coefficient of 1 inside the parentheses:

$$ 4\left(x^{2}-6 x\right) + 9\left(y^{2}+4 y\right) = -36 $$

3. Complete the squares.

Complete the perfect square trinomial by adding the square of half the linear coefficient inside the parentheses for the corresponding variable. Add these same values multiplied by their factor to the right side to keep the equation balanced:

$$ 4\left(x^{2}-6 x + 9\right) + 9\left(y^{2}+4 y + 4\right) = -36 + 4\left(9\right) + 9\left(4\right) $$

$$ 4\left(x - 3\right)^2 + 9\left(y + 2\right)^2 = 36 $$

4. Divide to equal 1.

Divide the entire equation by the value on the right side to get a 1 on that side, thus obtaining the standard form:

$$ \dfrac{\left(x - 3\right)^2}{9} + \dfrac{\left(y + 2\right)^2}{4} = 1 $$

Write the following ellipse in standard form: $ 25x^2+16y^2+150x-64y-111=0. $

Standard form of the ellipse

$$ \dfrac{\left(x + 3\right)^2}{16} + \dfrac{\left(y - 2\right)^2}{25} = 1 $$

Step-by-step solution

1. Identify and prepare the equation.

We start from the entered equation:

$$ 25x^2+16y^2+150x-64y-111=0 $$

Group the terms by variable and move the constant term to the right side of the equation:

$$ \left(25 x^{2}+150 x\right) + \left(16 y^{2}-64 y\right) = 111 $$

2. Factor out the leading coefficients.

Factor out the common factor from the quadratic terms so they have a coefficient of 1 inside the parentheses:

$$ 25\left(x^{2}+6 x\right) + 16\left(y^{2}-4 y\right) = 111 $$

3. Complete the squares.

$$ 25\left(x^{2}+6 x + 9\right) + 16\left(y^{2}-4 y + 4\right) = 111 + 25\left(9\right) + 16\left(4\right) $$

$$ 25\left(x + 3\right)^2 + 16\left(y - 2\right)^2 = 400 $$

4. Divide to equal 1.

Divide the entire equation by the value on the right side to get a 1 on that side, thus obtaining the standard form:

$$ \dfrac{\left(x + 3\right)^2}{16} + \dfrac{\left(y - 2\right)^2}{25} = 1 $$

Determine the standard equation of the ellipse given in general form $ 3x^2+5y^2-12x+30y+42=0. $

Standard equation of the ellipse

$$ \dfrac{\left(x - 2\right)^2}{5} + \dfrac{\left(y + 3\right)^2}{3} = 1 $$

Step-by-step solution

1. Identify and prepare the equation.

We start from the entered equation:

$$ 3x^2+5y^2-12x+30y+42=0 $$

Group the terms by variable and move the constant term to the right side of the equation:

$$ \left(3 x^{2}-12 x\right) + \left(5 y^{2}+30 y\right) = -42 $$

2. Factor out the leading coefficients.

Factor out the common factor from the quadratic terms so they have a coefficient of 1 inside the parentheses:

$$ 3\left(x^{2}-4 x\right) + 5\left(y^{2}+6 y\right) = -42 $$

3. Complete the squares.

$$ 3\left(x^{2}-4 x + 4\right) + 5\left(y^{2}+6 y + 9\right) = -42 + 3\left(4\right) + 5\left(9\right) $$

$$ 3\left(x - 2\right)^2 + 5\left(y + 3\right)^2 = 15 $$

4. Divide to equal 1.

Divide the entire equation by the value on the right side to get a 1 on that side, thus obtaining the standard form:

$$ \dfrac{\left(x - 2\right)^2}{5} + \dfrac{\left(y + 3\right)^2}{3} = 1 $$

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Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.