The correct option is
C 1aLet the tangents be at point
P(at21,2at1),QP(at22,2at2)
we know the point of intersection of tangents at any two given points
P(at21,2at1),Q(at22,2at2)is (at1.t2,2a(t1+t2))
since the tangents are mutually perpendicular and by property of tangents Mutually perpendicular tangents to a parabola meet on the directrixi.e
x=−a hence at1.t2=−aort1.t2=−1
Eqn of tangent to a parabola
y=mx+am
where m is slope of tangent
y2=4ax
2ym=4a
m=2ay
for P,m1=2a2at1=1t1
y=m1x+am1
y=xt1+at1
P1=puty=0
x=−at21
P1(−at21,0)
for Q,m2=2a2at2=1t2
y=xt2+at2
P2=puty=0
x=−at22
P2(−at22,0)
L(SP1)=a−(−at21)=a(1+t21)
L(SP2)=a−(−at22)=a(1+t22)
from 1 above
t1.t2=−1
1L(SP1)+1L(SP2)=1a(1+t21)+1a(1+t22)=1a⎛⎜
⎜⎝11+t21+11+1t21⎞⎟
⎟⎠=1a(1+t211+t22)=1a