Hyperbola

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)