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Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x²+9y²=36

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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.

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