The correct option is C 2√53
Let P(at21,2at1) and Q(at22,2at2)
For focal chord t1t2=−1
Let O be the vertex, then slope of PO=m1=2t1
And slope of QO=m2=2t2
tanθ=∣∣∣m1−m21+m1m2∣∣∣=∣∣∣2(t2−t1)t1t2+4∣∣∣
Let point of intersection of tangents at t1,t2 be R=(at1t2,a(t1+t2))
R lies on the directix x=−a and it also lies on line y=2x+a
∴ y=−a=a(t1+t2)
⇒ t1+t2=−1
(t2−t1)2=(t1+t2)2−4t1t2=5
t2−t1=±√5
tanθ=∣∣∣2√5−1+4∣∣∣=2√53