# Find the angle b/w the tangent drawn from the point (1,4) on the parabola y2=4x ?

1
by SURYASarin
angle between tangents

2014-12-17T22:13:45+05:30

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Slope of the tangents is dy/dx

2 y dy/dx = 4

m= slope=   dy/dx = 2/y

(x1,y1) and (x2,y2) are points where the 2 tangents touch the parabola.

we need to write the equation of tangent at (x1,y1) on the parabola and passing through (1,4) as
(y1 - 4) / (x1 - 1) = 2/y1
=>   y1² - 4 y1 = 2 x1 - 2 = 2 * (y1²/4) - 2 = y1²/2  - 2
=>   y1²/2 = 4 y1 - 2
=>    y1² -8 y1 + 4 = 0

y1 = (8 +- √(64-16))/2 =   4 +√12   and  y2 =  4-√12
x1 = y1²/4 =  7 +2√12        and     x2 =  7 - 2√12

There are two tangents, with above slopes.
m1 = tan Ф1 = 1/(2+√3) = 2-√3     , m2 = tanФ2 = 2+√3

Ф1 - Ф2 = angle between them.

tan (Ф1-Ф2) = (tanФ1 - tanФ2) / (1 + tanФ1 tan Ф2)
= 2√3 /2 = √3
angle between them is 60 degrees