Question

check whether the following statement are true or false. justify your answer with a short explanation or a counter example

the eccentricity of the conic x^2 + ky = 0, k>0 is 1/ (k+1)

Answer 1

y=-x^2/k is a parabola with fixed eccentricity=1. The parabola is an inverted U shape when k>0. With increasing values of k the U gets wider, but the eccentricity remains at 1. So the statement is false, since 1/(k+1) is only 1 when k=0; but k>0, by definition of k.