Identify the type of comic and write the equation in standard form

Question

X^2 - 4y^2 + 6x + 16y + 9 = 0

Answer 1

x^2 + 6x + 9 - y^2 + 16 y - 64 + 9 + 64 - 9 = 0

x^2 + 6x + 9 -(y^2 -16y +64) +64

complete the squares

add and subtract 9

add and subtract 64

(x+3)^2 - (y-8)^2 = -64

divide by 64

write 64 as 8^2

(x+3)^2/8^ - (y-8)^2/8^2 = -1

or eliminate the negative sign on the right

(y-8)^2/8^2 - (x+3)^2/8^2 = 1

The conic section is a rectangular hyperbola

It's rectangular because the asymptotes are perpendicular, y=x and y=-x

If you took a cone, with an inverted cone just below, joined together at a single point, and sliced it vertically, you would get two separate sections, which together form a hyperbola