Given equation: [tex]x^2 - 4y^2 - 2x + 16y - 31 = 0.[/tex]
Making it into groups of x and y terms.
( x^2-2x ) + (-4y^2 +16y ) -31 =0
Adding 31 on both sides.
( x^2-2x ) + (-4y^2 +16y ) =31.
Factoring of -4 from second group.
( x^2-2x ) -4 (y^2 - 4y ) = 31.
Completing the squares
(x^2 -2x +1) - 4(y^2 -4y +4)
We added 1 in first parenthesis and -4*4 = -16 in second parenthesis.
Adding 1 and -16 on right side too, we get
(x^2 -2x +1) - 4(y^2 -4y +4) = 31+1 -16
Making perfect squares
(x-1)^2 - 4(y-2)^2 = 16.
Dividing both sides by 16, we get
[tex]\frac{(x-1)^2}{16}-\frac{4(y-2)^2}{16}=\frac{16}{16}[/tex]
[tex]\frac{(x-1)^2}{16}-\frac{(y-2)^2}{4}=1[/tex].
