Circle

Let centre (h,k) and radius be r of circle S
∵S≡(x−h)2+(y−k)2=r2
Circle passes through origin
∵‌‌r2=h2+k2
Given, the line x=2 intersects the circle at two points let these points be ( 2,y1 ) and ( 2,y2 ) mid-points of these points is ( 2,k )
Now, the distance of (2,k) to (2,y1) is 2y1=k+2 and y2=k−2
( 2,k+2 ) and ( 2,k−2 ) are intersection point radius of the circle.
‌h2+k2=(h−2)2+(k−(k+2))2
‌h2+k2=h2+4−4h+4
‌k2=−4h+8⇒k2+4h=8
Taking locus, we get
y2+4x=8