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
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
Hello, Dan!
You're expected to know the general forms of the four conics.
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ must be equal.$\displaystyle 2x^2+Cy^2+Dx+Ey+F\:=\:0$ represents a conic.
State the values of $\displaystyle C$ for which each of the following are possible:
(a) a circle
. . $\displaystyle C \:=\:2$
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ are unequal, but have the same sign.(b) an ellipse
. . $\displaystyle C \:>\:0\:\text{ and }\:C \:\neq\: 2$
The equation has $\displaystyle x^2$ or $\displaystyle y^2$, but not both.(c) a parabola
. . $\displaystyle C \:=\:0$
The coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ must have opposite signs.(d) a hyperbola
. . $\displaystyle C \:<\:0$