Equation is a parabola with a vertical axis of symmetry

[gillgill]

Determine the restrictions on the constants A, C and E such that the following equation is a parabola with a vertical axis of symmetry.
Ax^2+Cy^2+x+Ey=0

a) A=0, C cannot=0, E cannot=0
b) A=0, C cannot=0, E=0
c) A cannot=0, C=0, E=0
d) A cannot=0, C=0, E cannot=0

i know that A cannot=0 and C=0...how about the x and y...what do they have to do with axis of symmetry?

 

[Galileo]

Here's how you should not do it (I think):
If C is zero you can immediately see you have the equation of a parabola with a vertical axis of symmetry. (This eliminates two answers)
Suppose c) is the answer. It's obvious you don't have a parabola.
So one answer remains.

BTW: A parab. with a vertical axis of symmetry has the form y=a(x-b)^2+c

 

[thepatientmental]

The correct answer is d) A cannot=0, C=0, E cannot=0.

In order for the equation to be a parabola with a vertical axis of symmetry, the coefficient of the x^2 term (A) cannot be equal to 0 because this would result in a linear equation. Additionally, the coefficient of the y^2 term (C) must be equal to 0 in order for the parabola to have a vertical axis of symmetry. This is because a parabola with a vertical axis of symmetry has a constant x-value for every y-value, meaning there is no variation in the x-value.

The constant term (E) cannot equal 0 because this would result in a parabola that passes through the origin, and a parabola with a vertical axis of symmetry does not pass through the origin. It must have a vertex that lies on the y-axis.

In summary, the restrictions on the constants A, C, and E for the equation to be a parabola with a vertical axis of symmetry are: A cannot equal 0, C must equal 0, and E cannot equal 0.