Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty. Enter the most specific answer.
x^2 + 2y^2 - 6x - 8y + 481 = 0
+128475 x^2 -2y^2 - 6x -8y = -481
x^2 - 6x - 2y^2 - 8y = -481 complete the square on x,y
x^2 - 6x + 9 - 2(y^2 -4y + 4) = -481 + 9 - 8
(x - 3)^2 -2 ( y -2)^2 = -480 divide through by 2
(x - 3)^2 /2 - (y - 2)^2 = - 240 rearrange as
(y -2)^2 - ( x - 3)^2/2 = 240 divide through by 240
(y - 2)^2/240 - (x - 3)^2/480 = 1
This is a hyperbola that opens up/down
