Completing the Square to Convert a Hyperbola Equation from General Form to Standard Form

Converting a Hyperbola: General Form to Standard Form

Completing the square is a useful technique for converting the equation of a hyperbola from general form to standard form. The general form of a hyperbola equation looks like:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

The standard form of a hyperbola equation looks like:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

, where $(h,k)$ is the center of the hyperbola, $a$ is the length of the transverse axis, and $b$ is the length of the conjugate axis.

Steps

###1: Isolate the $x$ and $y$ terms.

$$Ax^2 + Dx + Cy^2 + Ey = -Bxy - F$$

###2: Complete the Square for $x$

Group the $x$ terms together and complete the square. $$Ax^2 + Dx = A(x^2 + \frac{Dx}{A})$$

Add and subtract $(\frac{D}{2A})^2$ inside the parentheses. $$Ax^2 + Dx = A(x^2 + \frac{Dx}{A} + (\frac{D}{2A})^2 - (\frac{D}{2A})^2$$

Simplify inside the parenthesis: $$Ax^2 + Dx = A(x + \frac{D}{2A})^2 - \frac{D^2}{4A}$$

###3: Complete the Square for $y$ $$Cy^2 + Ey = C(y^2 + \frac{Ey}{C})$$

Add and subtract $(\frac{E}{2C})^2$ inside the parentheses. $$Cy^2 + Ey = C(y^2 + \frac{Ey}{C} + (\frac{E}{2C})^2 - (\frac{E}{2C})^2$$

Simplify inside the parenthesis: $$Cy^2 + Ey = C(y + \frac{E}{2C})^2 - \frac{E^2}{4C}$$

###4: Combine the Completed Squares

Substitute back into the original equation. $$A(x + \frac{D}{2A})^2 - \frac{D^2}{4A} + C(y + \frac{E}{2C})^2 - \frac{E^2}{4C} = -Bxy - F$$

###5: Move constant terms to the other side

$$A(x + \frac{D}{2A})^2 + C(y + \frac{E}{2C})^2 = \frac{D^2}{4A} + \frac{E^2}{4C} - Bxy - F$$

###6: Simplify! Use your mathematical tricks to simply and rearrange the equation so it looks like the standard form. $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

(Video) Example of Completing the Square for Hyperbolas

by Anusha Rahman

In this video by Anusha Rahman, you will learn how to complete the square to convert a hyperbola equation from general form to standard form.

Completing the Square: Hyperbolas

A hyperbola is a type of curve that has two separate parts. Its equation can sometimes look messy in general form:

$$Ax^2 + By^2 + Dx + Ey + F = 0$$

To graph or understand the hyperbola, we need to turn it into standard form by using a method called completing the square.

Example

Let's say you're given this equation: $$9x^2 - 4y^2 -54x + 8y = -137$$

Step 1: Group the $x$ terms and $y$ terms

$$(9x^2 -54x) - (4y^2 + 8y) = -137$$

Step 2: Factor out the number in front of each squared term

$$9(x^2 -6x) - 4(y^2 -2y) = -137$$

Step 3: Complete the square inside each parenthesis

• Half of $-6$ is $-3 \rightarrow (-3)^2 = 9$
• Half of $-2$ is $-1 \rightarrow (-1)^2 = 1$

So, add $9$ and $1$ inside the parentheses. But always remember to balance the equation!

You will add $81-4 = 77$ to the right side:

$$9(x-3)^2 - 4(y-1)^2 = -137 + 77$$

Step 4: Divide everything by $-60$ to make the equation equation to $1$:

$$\frac{(y-1)^2}{15} - \frac{(x-3)^2}{\frac{60}{9}} = 1$$