Sketch the following conic sections: 4^{2}-4y+8-4x^{2}=0

To determine: The radical form of $\sqrt[3]{64}$ in the rational exponent form.

Equation of a section plane in hyperbolic paraboloid
Find the equation of a plane passing through Ox and intersecting a hyperbolic paraboloid ${\displaystyle \frac{x^2}{p}-\frac{y^2}{q}=2z(p>0,q>0)}$ along a hyperbola with equal semi-axes.
My attempt: The equation of a plane passing through Ox is ${\displaystyle By+Cz=0}$. So ${\displaystyle z=\frac{-By}{C}}$. Now substitute ${\displaystyle z=\frac{-By}{C}}$ into the equation of hyperbolic paraboloid ${\displaystyle \frac{x^2}{p}-\frac{y^2}{q}=\frac{-2By}{C}}$ and transform to ${\displaystyle \frac{x^2}{p}-\frac{{(y-\frac{Bq}{C})}^2}{q}=-\frac{B^{2}q}{C^2}}$. What to do next? I don't understand how to find B and C if we assume ${\displaystyle \frac{x^2}{p}-\frac{{(y-\frac{Bq}{C})}^2}{q}=-\frac{B^{2}q}{C^2}}$ is a hyperbola with equal semi-axes.

Solve, a.Determine the conic section of the polar equation ${\displaystyle r=\frac{8}{2+2\sin \theta }}$ represents. b. Describe the location of a directrix from the focus located at the pole.

How do you find the eccentricity, directrix, focus and classify the conic section ${\displaystyle r=\frac{8}{4-1.6\sin \theta }}$?

A conical surface (an empty ice-cream cone) with surface charge density ? has height h.Radius of the top is R.Find potential difference between vertex & center of the top.

Identify the conic section given by ${\displaystyle y^2+2y=4x^2+3}$
Find its $\frac{\text{vertex}}{\text{vertices}}\text{and}\frac{\text{focus}}{\text{foci}}$