Question

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms, are ______.

विकल्प

• `6, 2sqrt(5)`

• `6, sqrt(5)`

• `4, 2sqrt(5)`

• `6 + 2sqrt(5), 2sqrt(5)`

Answer 1

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms, are `underlinebb(6 + 2sqrt(5), 2sqrt(5))`.

Explanation:

Given 2a = 6 and 2b = 4

∴ a = 3 and b = 2

Distance between two pins = SS' = 2ae

Here e = `sqrt(1 - b^2/a^2) = sqrt(1 - 4/9) = sqrt(5)/3`

∴ 2ae = `2 xx 3 xx sqrt(5)/3 = 2sqrt(5)`

Length of string = SPSS’

= SP + S’P + SS’

= 2a + 2ae

= `6 + 2sqrt(5)`

∴ Length of string = `6 + 2sqrt(5)` and distance between pins = `2sqrt(5)`