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

kkbisht
90 Points
2 years ago
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 $\theta$  b/w them is  tan $\theta$(m1-m2)/(1+m1m2)=1/2-(-1/2)/(1-1/2.1/2)  = 1/3/4 =4/3  => $\theta$$\tan^{-1}$(4/3)

kkbisht