1. finding values....

2x^2+Cy^2+Dx+Ey+F=0 represents a conic. state the values of C for which each of the following are possible

a circle

an ellipse

a parabola

a hyperbola

2. The general form of a second-degree quadratic in 2 variables is
$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

We have to check the expression $\displaystyle B^2 - 4AC$ to determine the type of conic.

If $\displaystyle B^2 - 4AC > 0$, then we have a hyperbola.

If $\displaystyle B^2 - 4AC = 0$, then we have a parabola.

If $\displaystyle B^2 - 4AC < 0$, then we have either a circle or an ellipse. If B = 0 and A = C, then we have a circle. Otherwise, it's an ellipse.

Now try your problem. B = 0 and A = 2...

01

3. Hello, Dan!

You're expected to know the general forms of the four conics.

$\displaystyle 2x^2+Cy^2+Dx+Ey+F\:=\:0$ represents a conic.
State the values of $\displaystyle C$ for which each of the following are possible:

(a) a circle
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ must be equal.
. . $\displaystyle C \:=\:2$

(b) an ellipse
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ are unequal, but have the same sign.
. . $\displaystyle C \:>\:0\:\text{ and }\:C \:\neq\: 2$

(c) a parabola
The equation has $\displaystyle x^2$ or $\displaystyle y^2$, but not both.
. . $\displaystyle C \:=\:0$

(d) a hyperbola
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ must have opposite signs.
. . $\displaystyle C \:<\:0$