If OT and ON are perpendiculars dropped from the origin to the tangent and normal to the curve
x = a sin 3 t, y = a cos 3 t at an arbitrary point, then
Text Solution
Verified by ExpertsD
(a, b, c)
At any point 't' on the given curve, we have 
=
= – cot t
The equations of the tangent and normal at 't' are
x cot t + y sin t – (a/2) sin 2t = 0 ….(i)
and , x sin t – y cos t – a cos 2t = 0 ….(ii)
∴ OT = 
= (a/2) sin 2t
⇒ 2. OT = a sin 2t
and, ON = 
= a cos 2t
Hence, 4 OT 2 + ON 2 = a 2 sin 2 2t + a 2 cos 2 2t = a 2
Length of the tangent at 't' = 
= 
= 
Length of the normal at 't'
= 
= 
= 
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