Hyperbolas
Nonstandard Form
If the equation is not given in standard form, it will need to be converted to standard form by completing the square. To identify an equation as a hyperbola, it will contain an x2 and y2 term where only one has a negative coefficient as in the example below.Example 4 Convert the following equation into standard form.
Step 1 . Identify the conic section.
This is a hyperbola because the x and y terms are both squared, and only one term has a negative coefficient, the y term.
Step 2 . Write equation in standard form.
(2x2+4x) + (-1y2+4y) = 4; Group the x terms, group the y terms
2(x2+2x) + -1(y2-4y) = 4; Factor the leading coefficient from each group
2(x2+2x+1) + -1(y2-4y+4) = 4 + (2?1) + (-1?4); Complete the square for x and y
2(x+1)2 + -1(y-2)2 = 2; Simplify
2(x2+2x) + -1(y2-4y) = 4; Factor the leading coefficient from each group
2(x2+2x+1) + -1(y2-4y+4) = 4 + (2?1) + (-1?4); Complete the square for x and y
2(x+1)2 + -1(y-2)2 = 2; Simplify
; Divide each side of the equation by 2 and simplify.