Question

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

Answer 1

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