Maths-
General
Easy
Question
Three normals drawn from any point to the parabola y2 = 4ax cut the line x = 2a in points whose ordinates are in arithmetical progression. Then the tangents of the angles which the normals make the axis of the parabola are in :
- A.P
- G.P
- H.P
- none
The correct answer is: G.P
To find the relation between tangents of angles.
The equation of normal in parametric form is
y=−tx+2at+at3
at^3+(2a−x)t−y=0
This is cubic in t
⇒t1+t2+t3=0
t1+t3=−t2...........(i)
At x=2a
y=at^3
So the ordinates at x=2a are 
Given 
substituting (i)
Slope of normal =tanϕ=t
(−tanϕ2)2=(−tanϕ1)(−tanϕ3)tan2ϕ2=tanϕ1tanϕ3
Clearly tanϕ1,tanϕ2,tanϕ3 are in G.P
Hence proved.
Hence, the tangent of the angles are in G.P.
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