How to Find the Equations of the Asymptotes of a Hyperbola

Step-by-Step Guide

1. 1

   Step 1: Write down the equation of the hyperbola in its standard form.

   We'll start with a simple example: a hyperbola with the center of its origin.
   
   For these hyperbolas, the standard form of the equation is x2/a2
   - y2/b2 = 1 for hyperbolas that extend right and left, or y2/b2
   - x2/a2 = 1 for hyperbolas that extend up and down.Remember, x and y are variables, while a and b are constants (ordinary numbers).
   
   Example 1: x2/9
   - y2/16 = 1 Some textbooks and teachers switch the position of a and b in these equations.Follow the equation closely so you understand what's going on.
   
   If you just memorize the equations you won't be prepared when you see a different notation.
2. 2

   Step 2: Set the equation equal to zero instead of one.

   This new equation represents both asymptotes, though it will take a little more work to separate them.Example 1: x2/9
   - y2/16 = 0 , Factor the left hand side of the equation into two products.
   
   Refresh your memory on factoring a quadratic if you need to, or follow along while we continue Example 1:
   We'll end up with an equation in the form (__ ± __)(__ ± __) =
   0.
   
   The first two terms need to multiply together to make x2/9, so take the square root and write it in those spaces: (x/3 ± __)(x/3 ± __) = 0 Similarly, take the square root of y2/16 and place it in the two remaining spaces: (x/3 ± y/4)(x/3 ± y/4) = 0 Since there are no other terms, write one plus sign and one minus sign so the other terms cancel when multiplied: (x/3 + y/4)(x/3
   - y/4) = 0 , To get the equations for the asymptotes, separate the two factors and solve in terms of y.
   
   Example 1:
   Since (x/3 + y/4)(x/3
   - y/4) = 0, we know x/3 + y/4 = 0 and x/3
   - y/4 = 0 Rewrite x/3 + y/4 = 0 → y/4 =
   - x/3 → y =
   - 4x/3 Rewrite x/3
   - y/4 = 0 →
   - y/4 =
   - x/3 → y = 4x/3 , We've just found the asymptotes for a hyperbola centered at the origin.
   
   A hyperbola centered at (h,k) has an equation in the form (x
   - h)2/a2
   - (y
   - k)2/b2 = 1, or in the form (y
   - k)2/b2
   - (x
   - h)2/a2 =
   1.
   
   You can solve these with exactly the same factoring method described above.
   
   Just leave the (x
   - h) and (y
   - k) terms intact until the last step.
   
   Example 2: (x
   - 3)2/4
   - (y + 1)2/25 = 1 Set this equal to 0 and factor to get: ((x
   - 3)/2 + (y + 1)/5)((x
   - 3)/2
   - (y + 1)/5) = 0 Separate each factor and solve to find the equations of the asymptotes: (x
   - 3)/2 + (y + 1)/5 = 0 → y =
   -5/2x + 13/2 ((x
   - 3)/2
   - (y + 1)/5) = 0 → y = 5/2x
   - 17/2
3. 3

   Step 3: Factor the new equation.

4. 4

   Step 4: Separate the factors and solve for y.

5. 5

   Step 5: Try the same process with a harder equation.

Detailed Guide

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