Consider the parabola $(x-1)^2 + (y-2)^2 = \frac{(12x - 5y + 3)^2}{169}$.
Step-by-Step Solution
Key Concept: The directrix is the locus of feet of perpendiculars from the focus to all tangents; the axis is perpendicular to the directrix through the focus.
The directrix of a parabola is perpendicular to its axis and passes through the focus. The locus of intersection points of perpendicular tangents forms the directrix, given by $12x - 5y + 3 = 0$. The axis is perpendicular to this directrix and passes through the focus $(1,2)$, with equation $5x + 12y - 29 = 0$. The minimum focal chord lies along the latus rectum $12x - 5y - 2 = 0$, and the tangent at vertex is $24x - 10y + 1 = 0$.
Correct Answer: [A-r] [B-s] [C-p] [D-q]