Question

Answer the following:

Find the

1. lengths of the principal axes
2. co-ordinates of the foci
3. equations of directrices
4. length of the latus rectum
5. distance between foci
6. distance between directrices of the ellipse:

`x^2/25 + y^2/9` = 1

Answer 1

Given equation of the ellipse is `x^2/25 + y^2/9` = 1.

Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get

a² = 25 and b² = 9

∴ a = 5 and b = 3

Since a > b,

X-axis is the major axis and Y-axis is the minor axis.

i. Length of major axis = 2a = 2(5) = 10

Length of minor axis = 2b = 2(3) = 6

∴ Lengths of the principal axes are 10 and 6.

ii. We know that e = `sqrt("a"^2 - "b"^2)/"a"`

∴ e = `sqrt(25- 9)/5`

= `sqrt(16)/5`

= `4/5`

Co-ordinates of the foci are S(ae, 0) and S'(–ae, 0),

i.e., `"S"(5(4/5),0)` and `"S'"(-5(4/5),0)`,

i.e., S(4, 0) and S'(–4, 0)

iii. Equations of the directrices are x = `± "a"/"e"`,

i.e., x = `± 5/(4/5)`, i.e., x = `± 25/4`

iv. Length of latus rectum = `(2"b"^2)/"a" = (2(3)^2)/5 = 18/5`

v. Distance between foci = 2ae = `2(5)(4/5)` = 8

vi. Distance between directrices = `(2"a")/"e"`

= `(2(5))/(4/5)`

= `25/2`