Find the lengths of the major and minor axes, coordinates of the vertices and the foci, the eccentricity and length of the latus rectum of the ellipse:
`x^(2)/4+y^(2)/36 = 1.`

Explanation

Given equation is `x^(2)/4+y^(2)/36 = 1`.
This is of the form ` x^(2)/b^(2) +y^(2)/a^(2) =1," where " a^(2) gt b^(2)`.
So, it is an equation of a vertical ellipse.
Now, `(b^(2) = 4 and a^(2) = 36) rArr (b=2 and a = 6)`.
`:. c = sqrt(a^(2)-b^(2)) = sqrt(36 -4) = sqrt(32) = 4 sqrt2`.
Thus, ` a =6, b = 2 and c = 4sqrt2`.
(i) Length of the major axis = `2a = (2xx6) ` units = 12 units.
Length of the minor axis = `2b = (2xx2) ` units = 4 units.
(ii) Coordinates of its vertices are A(0, -a) and B(0,a), i.e., A(0, -6) and B(0, 6).
(iii) Coordinates of its foci are `F_(1)(0,-c) and F_(2)(0,c), i.e., F_(1)(0, -4)sqrt2) and F_(2)(0, 4sqrt2)`.
(iv) Eccentricity, `e=c/a = (4sqrt2)/6 = (2sqrt2)/3`.
(v) Length of the latus rectum = `(2b^(2))/a = ((2xx4))/6 " units " = 4/3` units.