Equation of the normal at t is
tx + y = 2at + at3
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Tangent at (at2, 2at) is ty = x + at2 . Thus the slope of the tangent at t is 1/t. Hence, the equation of the normal at (at2, 2at) is
y – 2at = -t(x – at2)
⇒ tx + y = 2at + at3