Conic sections

The parabola $x^2 = 12y$ has its focus at $(0, p)$. What is $p$?

For every point $P$ on the ellipse $\dfrac{x^2}{49} + \dfrac{y^2}{24} = 1$, what is $PF_1 + PF_2$, the sum of the distances to the two foci?

The ellipse $16x^2 + 25y^2 = 400$ has foci at $(\pm c, 0)$. Find $c$.

The hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$ has foci at $(\pm c, 0)$. Find $c$.

The equation $2x^2 + 3xy + y^2 + x - 5 = 0$ describes a non-degenerate conic. Which one?

The equation $x^2 + y^2 - 6x + 8y = 0$ describes a circle. What is its radius?

Which of these equations describe something other than an ellipse, parabola or hyperbola? (Choose all that apply.)

In the proof of the reflective property of the parabola, $D$ is the foot of the perpendicular from $P$ to the directrix, and the tangent $t$ at $P$ is the perpendicular bisector of $FD$. Why does this show that the reflected ray passes through $F$?

Which Lean statement correctly says that $(x, y)$ is equidistant from the focus $(0, p)$ and the directrix $y = -p$ exactly when $x^2 = 4py$?

Write ellipse_foci(a, b) returning the two foci of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a, b > 0$) as a sorted list of two (x, y) tuples of floats. Careful: the major axis is horizontal only when $a \ge b$!