Question

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.`(x^2)/(16)+(y^2)/9=1`

Answer 1

Given equation is ` x^(2)/16+y^(2)/9 =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) =16 and b^(2) = 9 rArr (a=4 and b=3)`.
` :. C = sqrt(a^(2)-b^(2)) = sqrt(16-9) = sqrt7`.
Thus, ` a =4, b=3 and c = sqrt7`.
(i) Length of the major axis =`2a=(2xx4)` units = 8 units.
Length of the minor axis = `2b = (2 xx3)` units = 6 units.
(ii) Coordinates of the vertices are `A(-a, 0) and B(a, 0) , i.e., A(-4, 0) and B(4,0).`
(iii) Coordinates of the foci are `F_(1)(-c,0) and F_(2)(c,0),i.e., F_(1) (-sqrt7, 0) and F_(2)(sqrt7, 0)`.
(iv) Eccentricity , ` e = c/a = sqrt7/4`.
(v) Length of the latus rectum = `(2b^(2))/a = ((2xx9))/3 " units " =9/2 ` units.