Find the center, foci, vertices, asymptotes, and radius, if necessary, of the conic sections in the equation: displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}

To determine: Simplify the given expression

We need to verify theorem 5 Focus-Directrix definition: ${\displaystyle 0<e<1\text{and}c=d(e-2-1)}$

To determine. To fill: the blank in given sentence.

How to find the position on ellipse (or hyperbola) arc if we know it's euclidean distance from given point and direction of movement?

Convert the following equations to its standard form and identify the vertex and focus.
${\displaystyle 3y^2+8x+24y+40=0}$

Where is the vertical directrix of a conic if the denominator of its polar equation has the form ${\displaystyle 1-e\cos q}$?

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.