The length of the chord of the parabola `x^(2) = 4y` having equations `x - sqrt(2) y + 4 sqrt(2) = 0` is

Options

A. `8 sqrt(2)`
B. `2 sqrt(11)`
C. `3 sqrt(2)`
D. `6 sqrt(3)`

Answer

D

Explanation

Given equation of parabola is `x^(2) = 4y` …..(i)
and the chord is `x - sqrt(2) y + 4 sqrt(2) = 0`
From Eqs. (i) and (ii) we have
`[sqrt(2) (y - 4)]^(2) = 4y`
`implies 2(y - 4)^(2) = 4y`
`implies (y - 4)^(2) = 2y`
`implies y^(2) - 8y + 16 = 2y`
`implies y^(2) - 10y + 16 = 0`

Let the roots Eq. (iii) be `y_(1)` and `y_(2`)
Then, `y_(1) + y_(2) = 10` and `y_(1)y_(2) = 16`
Again from Eqs. (i) and (ii) we have
`x^(2) = 4 [(x)/(sqrt(2)) + 4]`
`implies x^(2) - 2 sqrt(2x) - 16 = 0`
Let the roots of Eq. (v) be `x_(1)` and `x_(2)`
Then, `x_(1) + x_(2) - 2 sqrt(2)`
and `x_(1)x_(2) = - 16`
Clearly, length of the chord AB
`= sqrt((x_(1) - x_(2))^(2) + (y_(1) - y_(2))^(2))`
`= sqrt((x_(1) + x_(2))^(2) - 4x_(1)x_(2) + (y_(1) + y_(2))^(2) - 4 y_(1)y_(2))`
`[:' (a - b)^(2) = (a + b)^(2) - 4 ab]`
`= sqrt(8 + 64 + 100 - 64)`
`= sqrt(108) = 6 sqrt(3)`
[from Eqs. (iv) and (vi)]