Answer
Parabola
Work Step by Step
$x+3y=2{{y}^{2}}-1$
It has only one variable that is squared, so the graph cannot be a circle, an ellipse, or a hyperbola.
Rewrite the equation as shown below,
$2{{y}^{2}}-x-3y-1=0$
Compare the equation $2{{y}^{2}}-x-3y-1=0$ with the standard equation of a conic section $A{{x}^{2}}+B{{y}^{2}}+Cxy+Dx+Ey+F=0$,
Here,
$A=0,B=2$
The conditions are $A=0$ or $B=0$, but not both.
Thus, the provided equation $2{{y}^{2}}-x-3y-1=0$ has $B=0$.
Now rewrite the provided equation $x+3y=2{{y}^{2}}-1$ in the standard form of a parabola $x=a{{y}^{2}}+by+c$,
$\begin{align}
& x+3y=2{{y}^{2}}-1 \\
& x=2{{y}^{2}}-3y-1
\end{align}$
Therefore, the graph of the equation is a parabola.
