Correct Answer - C
We know tangents to `y^(2) = 4 ax` is `y = mx + (1)/(m)`
`:.` Tangent to `y^(2) = 4x` is `y = mx + (1)/(m)`
Since tangent passes through (1,4)
`:. 4 = m + (1)/(m)`
`implies m^(2) - 4m = 4` (whose roots are `m_(1)` and `m_(2)`)
`:. m_(1) + m_(2) = 4` and `m_(1) m_(2) = 1`
and `|m_(1) - m_(2)| = sqrt((m_(1) + m_(2))^(2) - 4m_(1)m_(2))`
`= sqrt(12) = 2 sqrt(3)`
Thus, angle between tangents
`tan theta = |(m_(2) - m_(1))/(1 + m_(1) m_(2))| = (2 sqrt(3))/(1 + 1) = sqrt(3) implies theta = (pi)/(3)`
