A particle is projected from origin with speed `u`. Find the minimum value of `u` if the particle passes through a poiny `P (a, sqrt(3)a)`.

Explanation

Let `theta` be an angle of projection. The equation of trajectory of projectile
`y = x tan theta - (g x^(2))/(2u^(2) cos^(2) theta)`
`2u^(2)y = 2u^(2) x tan theta - gx^(2)sec^(2) theta`
`2 u^(2)sqrt(3)a = 2u^(2)a tan theta - g a^(2) (1 tan^(2) theta)`
`g a^(2) tan^(2) theta - 2 u^(2) a tan theta + (ga^(2) + 2u^(2)sqrt(3) a) = 0`
if the particle passes through `P`, for this `tan theta` should be real, for `tan theta` to be real, discriminant
`[(- 2u^(2) a)^(2) - 4 g a^(2)(g a^(2) + 2 sqrt(3) u^(2) a)] ge 0`
`u^(4) - g^(2) a^(2) - 2 sqrt(3) u^(2) g a ge 0`
`u^(4) - 2sqrt(3) u^(2) g a + 3 g^(2) a^(2) ge (g^(2) a^(2) + 3 g^(2) a^(2))`
`(u^(2) - sqrt(3) g a^(2)) ge (2 g a)^(2)`
`u^(2) - sqrt(3) g a ge 2 ga`
`u ge sqrt((2 + sqrt(3))g a)`
`u_(min) = sqrt((2 + sqrt(3)) ga)`
OR
`y = x tan theta - (g x^(2))/(2u^(2) cos^(2) theta)`
`sqrt(3) a = a tan theta - (ga^(2) sec^(2) theta)/(2u^(2))` (i)
`2sqrt(3) u^(2) a = 2a u^(2) tan theta = - g a^(2) sec^(2) theta`
For `u` to be minimum `(d u)/(d theta) = 0`
`2sqrt(3) a. 2u (d u)/(d theta) = 2a [2u (d u)/(d theta)tan theta + u^(2) sec^(2) theta]`
`- g a^(2) 2sec theta. sec theta tan theta = 0`
`0 = 2a u^(2) sec^(2) theta - 2 ga^(2) sec^(2) theta tan theta`
`= u^(2) - g a tan theta rArr tan theta = (u^(2))/(g a)` in (i)
`sqrt(3) a = a tan theta = - (g a^(2))/(2u^(2))(1 + tan^(2) theta)`
`= a. (u^(2))/(g a) - (g a^(2))/(2 u^(2))(1 + (u^(4))/(g^(2)a^(2)))`
Solving `u = sqrt((2 + sqrt(3)) ga)`