The length of the major axis of the ellipse $(5x - 10)^2 + (5y + 15)^2 = \frac{(3x - 4y + 7)^2}{4}$ is $A$. Find $[A]$. ($[|$ represents greatest integer function$)$
Step-by-Step Solution
Key Concept: A parabola is the locus of points equidistant from a focus and directrix; use the focal distance formula to find the parameter.
The equation $\sqrt{(x-2)^2 + (y+3)^2} = \frac{1}{2}|3x - 4y + 7|$ represents a parabola with focus at $(2, -3)$, directrix $3x - 4y + 7 = 0$, and eccentricity $e = \frac{1}{2}$. Using the focal property that the distance from focus to directrix equals $\frac{a}{e} - a$, we get $5 = \frac{3a}{2}$, yielding $a = \frac{10}{3}$ and $A = \frac{20}{3}$.
Correct Answer: 6