Find the locus of the mid-point of the chords of the parabola y^2=4ax such that tangent at the extremities of the chords are prependicular.

the Vertex is O(0.0), which is one end of the chord. Let the other end be a varaible point P given by (at2,2at).Let M(p,q) be the midpoint of the chord OP. Midpoint of OP is (at2/2,at).So,p=at22andq=atNow we have to eliminate t and get the relation between p and q to get the locus.So,t=qaSubstitute this in the equation of p, and we will get:p=a2(qa)2So we have,q2=2ap