4x2 + 9y2 = 1 can be written as x2/(1/4) + 9y2/(1/9) = 1
For general form of ellipse:
x2/a2 + y2/b2 = 1 ….(1)
On comparing given equation with (1), we get
a = 1/2 and b = 1/3
Then,
c2 = a2 – b2
c2 = 1/4 – 1/9 = 5/36
or c = √5/6
Now,
(i) Lengths of major and minor axes:
Major axis: 2a = 1 units
Minor axis : 2b = 2/3 units
(ii) Coordinates of the vertices: (±a, 0) = (±1/2, 0)
(iii) Coordinates of the foci: (±c, 0) = (±√5/6, 0)
(iv) Eccentricity: e = c/a = √5/3
(v) Length of the latus rectum: 2b2/a = 4/9 units