Problem 1: Determine the standard equation and key geometric properties for a given set of coordinates.

Problem 2: Convert the general equation $4x^2 - 8x + 16y - 20 = 0$ into standard form.

Problem 3: Find the focus, directrix, and axis of symmetry for the conic section defined by $x^2 = -8y$.

Case Study 1: A parabolic reflector is designed such that its cross-section is a parabola. If the diameter of the opening is $24$ cm and the depth is $6$ cm, where should the light source be placed to ensure parallel beams?

Case Study 2: Calculate the length of the latus rectum for the curve defined by $y^2 - 10x - 4y + 24 = 0$.