The correct option is
D 8 a2Let (at2,2at1),(at22,2at2),(at23,2at3) be coordinates of the point P, Q, R respecting which lie on the curve y2=4ax.
Now, equation of the normal to the curve at P is,
y−2at1=−2at12a(x−at21) ……….(i)
R is a point which lies at the normal at P.
∴ Substituting y=2at3, x=at23 in equation (1)
2at3−2at1=−t1(at23−at21)
2a(t3−t1)=−at1(t23−t21)
2(t3−t1)=−t1(t3−t1)(t3+t1)
2=−t1(t3+t1)
⇒2=−t1t3−t21 ……….(ii)
Now, the equation of the normal to the curve at Q is
y−2at2=−2at22a(x−at22) ………..(iii)
R is a point which lies at the normal at Q.
Substituting y=2at3,x=at23 in equation (iii)
2at3−2at2=−t2(at23−at22)
2a(t3−t2)=−at2(t3−t2)(t3+t2)
2=−t2(t3+t2)
⇒2=−t2t3−t22 ……..(iv)
Multiplying equation (ii) with t2,
2t2=−t1t2t3−t21t2 ………..(v)
Multiplying equation (iv) with t1,
2t1=−t1t2t3−t22t1 …….(vi)
Subtracting (v) and (vi), we get
2t2−2t1=−t21t2+t22t1
2(t2−t1)=−t1t2(t1−t2)
2(t2−t1)=t1t2(t2−t1)
⇒t1t2=2
Now, the product of ordinates of P and Q=2at1⋅2at2
=4a2t1t2
=4a2(2)
=8a2.