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The angle between the tangents drawn fro...

The angle between the tangents drawn from the point `(1,4)` to the parabola `y^2=4x` is

A

`(pi)/(6)`

B

`(pi)/(4)`

C

`(pi)/(3)`

D

`(pi)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
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)`
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