find the angle between the tangents drawn from the point (4,0) to the parabola y^2+4x=0

We know that a line y=mx+c is  tangent to the standard parabola  y²=4ax if c=a/m therefore the line y=mx+a/m is always a tangent to the parabola y²=4ax. Here the equation of parabola is given as y²= -4x where a= -1. Hence the equation of tangent to this parabola is y=mx-1/m and it is given that  this tangent passes through the point (4,0) therefore 0=4m-1/m  => 4m²=1 =. m²=1/4=> m=+-½ .Therefore the two tangents can be drawn to the parabola from the point (4,0)   with slopes m₁=-1/2 amd m₂=1/2

Therefore the angle   b/w them is  tan = (m₁-m₂)/(1+m₁m₂)=1/2-(-1/2)/(1-1/2.1/2)  = 1/_3/4 =4/3  => = (4/3)

 

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