If two tangents drawn from the point `(alpha,beta)` to the parabola `y^2=4x` are such that the slope of one tangent is double of the other, then prove that `alpha=2/9beta^2dot`

Options

A. `9 beta = 2alpha^(2)`
B. `9 alpha = 2 beta^(2)`
C. `2 alpha = 9 beta^(2)`
D. None of these

Answer

B

Explanation

Equation of tangent to the parabola `y^(2) = 4x` is y = mx `+ (1)/(m) " " ….(1)`
it passes through `(alpha , beta)`
`therefore beta = m alpha + (1)/(m)`
`implies m^(2) alpha - mbeta + 1 = 0`
let it roots are `m_(1)` and `m_(2)`
according to condition if `m_(1)` = m then `m_(2) = 2m `
sum of roots `m_(1) +m_(2) = m + 2m = (beta)/(alpha)`
`implies 3m = (beta)/(alpha)`
`implies m = (beta)/(3 alpha) " " .......(2)`
Product of roots `m_(1).m_(2)= (m) (2m) = (1)/(alpha)`
`implies 2m^(2) = (1)/(alpha)`
from (1)
`(2.beta^(2))/(9 alpha^(2)) = (1)/(alpha) implies 2 beta ^(2) = 9 alpha `