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Equation of tangent in slope form is y=mx + a/m :-1This equation represents 2 lines.Let the external point be (g,f)Putting this in equation 1, we get f=(g)m +a/(m)Solving we get, (g)(m^2) -(f)(m) +a=0. :-2 Given that x1+x2=45°x1 and x2 are angles of tangents with x- axis thus tan of the angles represents their slopes. tan(x1+x2)=tan 45° ( tan x1 +tan x2)/[1-(tan x1 tan x2)]=1. :-3From the quadratic equation 2 we get tan x1 + tan x2=(f)/g. :-4 tan x1 tan x2=a/g. :-5 Putting 4,5 in 3 we have f=g-a From equation of parabola y^2=4x we have a=1 f=g-1 g-f-1=0The locus of all such points is x-y-1=0Hope it will help you.