Let the mid-point of the chord is (h1k). Then, chord through mid - point (h,k) is T=S1 xh−yk=h2−k2 ...(i) Now, this is also a tangent of y2=8x The equation of the tangent of slope m to the parabola y2=8x is given by Tangent : y=mx+
2
m
⇒m2x−my=−2 ...(ii) Eqs. (i) and (ii) are coincide ∴
h
m2
=
−k
−m
=
h2−k2
−2
⇒h=km ⇒m=
h
k
∴
k2
h
=
h2−k2
−2
⇒−2k2=h3−hk2 ⇒h3=k2(h−2) Therefore, locus of mid-point of the chords, x3=y2(x−2)