The correct option is
B y2=2(x−4)Let the equation of the chord
OP be
y=mx and then,
Equation of the chord OQ will be =y=−1mx and
P is the point of intersection of y=mx and y2=4x is (4m2,4m) and
Q is the point of intersection of y=−xm and y2=4x is (4m2,−4m)
Now, the equation of PQ is y+4m= 4m+4m4m2−4m2(x−4m2)
⇒y+4m=m1−m2(x−4m2)
⇒(1−m2)y+4m−4m3=mx−4m3
⇒ mx−(1−m2)y−4m=0
This line meets x−axis, where y=0
i.e, x=4 ⇒OL=4 which is constant as independent of m.
Again let (h,k) be the mid-point of PQ, then
h=4m2+4m22 and k =4m−4m2
⇒h=2(m2+1m2) and k=2(1m−m)
⇒h=2((m−1m)2+2) and k=2(1m−m)
Eliminating m, we get
2h=k2+8
y2=2(x−4) is required equation of locus.