Question

is 2x^2+2y^2=8x-16y-32 is it a circle, parabola, hyperbola or ellipse?

Answer 1

Get all the x's and y's on one side. Complete the square, then look at your result.
So we have a factor of 2 in EVERY term, so divide it out and move everything giving?
\[x^2+y^2-4x+8y=-16\]
To complete the square, divide the term multiplying the linear terms (-4x and 8y) so -4 and 8. Divide them by 2 and then square them, then you add and subtract that number:
\[(x^2-4x+4-4)+(y^2+8y+16-16)=-16\]
Now factor the "perfect squares" we have just made:
\[(x-2)^2-4+(y+4)^2-16=-16\]
Adding over the 4 and 16 gives:
\[(x-2)^2+(y+4)^2=4\]
Thus we have a circle.

Answer 2

(A circle is just an ellipse but the radii (major and minor axes) are the same length.) All circles are ellipses but all ellipses are not circles. Its only a one way implication.