Question

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:
`4x^(2)+9y^(2)=144.`

Answer 1

The given equation may be written as
`x^(2)/36 +y^(2)/16 =1`.
This is of the form `x^(2)/a^(2) + y^(2)/b^(2) = 1," where " a^(2) gt b^(2)`.
So, it is an equation of a horizontal ellipse.
Now, `(a^(2) = 36 and b^(2) = 16 ) rArr (a=6 and b = 4)`.
` :. c = sqrt(a^(2)=b^(2)) = sqrt(36-16) = sqrt(20) = 2sqrt5`.
Thus, a = 6, b = 4 and ` c = 2sqrt5 .`
(i) Length of the major axis = `2a=(2xx6)` units = 12 units.
Length of the minor axis = ` 2b = (2 xx4) ` units = 8 units.
(ii) Coordinates of the vertices are A(-a, 0) and B(a,0) , i.e., A(-6,0) and B(6,0).
(iii) Coordinates of the foci are `F_(1)(-c, 0) and F_(2)(c,0), i.e., F_(1)(-2sqrt5, 0) and F_(2)(2sqrt5, 0)`.
(iv) Eccentricity, ` e = c/a = (2sqrt5)/6 = sqrt5/3`.
(v) Length of the latus rectum = `(2b^(2))/a = ((2xx16))/6 " units " = 16/3` units.