Ellipse

Concept:For an ellipse with horizontal major axis, the center $(h,k)$, focus $(h\pm c,k)$, vertex $(h\pm a,k)$.The relation $c^2 = a^2 - b^2$ holds.Length of latus rectum is $\frac{2b^2}{a}$.Explanation:Given center $(1,-2)$, focus $(3,-2)$, vertex $(5,-2)$.All have same $y$ coordinate, so major axis is horizontal.Semi-major axis $a$ = distance from center to vertex = $|5-1| = 4$.Distance from center to focus $c = |3-1| = 2$.Using $c^2 = a^2 - b^2$: $2^2 = 4^2 - b^2 \Rightarrow 4 = 16 - b^2 \Rightarrow b^2 = 12$.Length of latus rectum = $\frac{2b^2}{a} = \frac{2 \times 12}{4} = \frac{24}{4} = 6$.Answer:6 (Option A)