When does Ax2 + Cy^2 + Dx + Ey + F = 0 represent a parabola

[gillgill]

1. Determine the restrictions on the constants A, C and D so that Ax^2+Cy^2+Dx+Ey+F=0 represents a parabola that opens to the left.
I know that A=0, how do u know if CD>0 or CD<0...??

2. The equation Ax^2+By^2+Cy=0 represents an ellipse. If 0<A<B<C, then the vertices of the ellipse are on
a line parallel to the y-axis or a line parallel to the x-axis?

3. Determine the restrictions on the constants, A, C and E such that the following equation is a parbola with a vertical axis of symmetry. Ax^2+Cy^2+x+Ey=0
i know that A can't =0 and C=0...
how can u tell if E=0 or not...?

Thanks.

 

[Berislav]

1. What is the most general form of the equation for a parabola? Write it implicitaly and compare to the restrictions you have.
What condition must be met for the parabola to open to the left?

2. What are the vertices of an ellipse?

3. Again the same as in 1. What does it mean that it's axis of symmetry is vertical? Imagine such a parabola.

 

[thepatientmental]

1. To determine if the equation represents a parabola that opens to the left, we need to look at the coefficient of x^2, which is A. For a parabola that opens to the left, A must be negative. This means that the restriction on A is A < 0.

To determine the restrictions on C and D, we can look at the discriminant of the quadratic equation formed by the x terms. The discriminant is given by D = B^2-4AC. For a parabola that opens to the left, the discriminant must be positive, so we have B^2-4AC > 0. Since B is not present in the given equation, this means that C must be positive, so the restriction on C is C > 0.

2. The given equation Ax^2+By^2+Cy=0 represents an ellipse. If 0<A<B<C, then the vertices of the ellipse will lie on a line parallel to the y-axis. This is because the larger coefficient, C, is associated with the y^2 term, indicating that the ellipse is stretched more in the y-direction.

3. To determine if the equation represents a parabola with a vertical axis of symmetry, we need to look at the coefficient of y^2, which is C. For a vertical axis of symmetry, C must be negative. This means that the restriction on C is C < 0.

To determine the restrictions on A and E, we can look at the discriminant of the quadratic equation formed by the y terms. The discriminant is given by D = B^2-4AC. For a parabola with a vertical axis of symmetry, the discriminant must be positive, so we have B^2-4AC > 0. Since B is not present in the given equation, this means that A must be positive, so the restriction on A is A > 0.

As for the constant E, we can look at the coefficient of y, which is E. If E = 0, then the parabola will not have a vertical axis of symmetry. Therefore, the restriction on E is E ≠ 0.