The correct option is D Focus has the co-ordinates (a,0)
Any point on the parabola is
P(at2,2at)
Therefore, the midpoint of S(a,0) and P(at2,2at) is,
R(a+at22,at)≡(h,k)
⇒h=a+at22,k=at
Eliminating t,
2x=a(1+y2a2)=a+y2a
⇒2ax=a2+y2
⇒y2=2a(x−a2)
It is a parabola with vertex at (a2,0) and latus rectum 2a.
The directrix is
x−a2=−a2
⇒x=0
The focus is
x−a2=a2
⇒x=a
Therefore the focus is (a,0)