Question

let, ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 be a curve. If h^2 = ab, then prove that the curve is a parabola.

Answer 1

for the curve to be a parabola, either a or b must be 0, since h^2 = ab then h is 0
Let b=0
-->ax^2 +2gx +2fy +c = 0
this is a vertical parabola

Answer 2

This is just a special case. for given condition, the curve must represent parabola with focus at any point and any directrix .. anyway thanks for try