Final answer:
The ellipse 4x²+9y²=36 has its foci at (±√5, 0), vertices at (±3, 0), a major axis of 6, a minor axis of 4, an eccentricity of √5/3, and a latus rectum length of approximately 2.67.
Step-by-step explanation:
To find the characteristics of the ellipse defined by the equation 4x²+9y²=36, we first rewrite the equation in standard form:
\(rac{x²}{9}+\frac{y²}{4}=1\)
From the equation, we see that a²=9 and b²=4, where a is the semi-major axis and b is the semi-minor axis. This gives us a=3 and b=2.
The major axis is 2a = 6, and the minor axis is 2b = 4.
The foci are located at a distance c from the center, where c=√(a²-b²). Calculating that gives c=√(9-4)=√5. So, the coordinates of the foci are (±√5, 0).
The vertices are located at (±a, 0) = (± 3, 0).
The eccentricity e of the ellipse is given by e = c/a. In this case, e = √5/3.
The length of the latus rectum is given by 2b²/a, which is 2 × 4/3, or approximately 2.67.