Hyperbolas
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.
; Divide each side of the equation by 2 and simplify.
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.