The equation of normal in parametric form is
y=−tx+2at+at3
at3+(2a−x)t−y=0
This is cubic in t
⇒t1+t2+t3=0t1+t3=−t2...........(i)
At x=2a
y=at3
So the ordinates at x=2a are at31,at32,at33
Given 2at32=at31+at33
2t32=t31+t332t32=(t1+t3)(t21+t23−t1t3)
substituting (i)
2t32=−t2(t21+t23−t1t3)−2t22=t21+t23−t1t3−2t22=(t1+t3)2−3t1t3−2t22=(t2)2−3t1t3−3t22=−3t1t3t22=t1t3
Slope of normal =tanϕ=t
(−tanϕ2)2=(−tanϕ1)(−tanϕ3)tan2ϕ2=tanϕ1tanϕ3
Clearly tanϕ1,tanϕ2,tanϕ3 are in G.P
Hence proved.