Question

The slope of the normal to the curve x=1/t ,y=2t at t=2.​

Answer 1

Answer:

It x=t

2

and y=2t then equation of normal at t=1 is -

X+Y-3=0

Answer 2

$\large{\underline{\underline{\textsf{\textbf{\purple{Given:}}}}}}$

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Curve Is :

• $\sf{x\:=\:{\dfrac{1}{t}}}$

• $\sf{y\:=\:2t}$

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$\large{\underline{\underline{\textsf{\textbf{\purple{Find:}}}}}}$

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• Slope of the normal to the curve at t = 2.

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$\large{\underline{\underline{\textsf{\textbf{\purple{Solution:}}}}}}$

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• $\sf{x\:=\:t}^{-1}$

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$\:\:\:\:\:\:\::\:\Longrightarrow\:\sf{\dfrac{dx}{dt}}\:{=\:-\:t}^{-2}$

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• $\sf{y\:=\:2t}$

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$\:\:\:\:\:\:\::\:\Longrightarrow\:\sf{\dfrac{dy}{dt}}\:{=\:2}$

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Now,

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$\:\:\:\:\:\:\::\:\Longrightarrow\:\sf{\dfrac{dy}{dx}}\:=\:\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

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$\:\:\:\:\:\:\::\:\Longrightarrow\:\sf{\dfrac{dy}{dx}}\:=\:{-2t}^{2}$

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Then,

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• Slope of the normal to the curve at t = 2.

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$\:\:\:\:\:\:\:\:\dashrightarrow\:\sf{m\:=\:\bigg (\:{\dfrac{dy}{dx}}\:\bigg )\:_t\:_=\:_2}$

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$\:\:\:\:\:\:\:\:\dashrightarrow\:\sf{m\:=\:-\:2\:{(2)}^{2}}$

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$\:\:\:\:\:\:\:\:\dashrightarrow\:\sf{m\:=-8}$

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• Slope of Normal :

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{-1}{m}}$

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$\:\:\:\:\:\:\:\leadsto\:\sf{8}$

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$\:\:\:\:\:\:\:\therefore\:{\underline{\sf{Slope \:of\: the\: Normal\: to \:the\: Curve\: at\: t \:=\: 2\: is \:{\textsf{\textbf{8}}}}}}.$

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