Find the lengths of subtangent, subnormal at a point '$t$' on the curve $x = a(Cos\,t + t\,Sin\,t)$, $y = a(Sin\,t - t\,Cos\,t)$.

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> [!TLDR]
> For a parametric curve, find $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ at parameter $t$, then apply the standard formulas Subtangent $=\dfrac{y}{dy/dx}$, Subnormal $=y\cdot\dfrac{dy}{dx}$.
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> Given $x = a(\cos t + t\sin t)$, $y = a(\sin t - t\cos t)$.
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> **Differentiate w.r.t. $t$:**
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> $\dfrac{dx}{dt} = a(-\sin t + \sin t + t\cos t) = a\,t\cos t$
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> $\dfrac{dy}{dt} = a(\cos t - \cos t + t\sin t) = a\,t\sin t$
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> **Slope:**
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> $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{a\,t\sin t}{a\,t\cos t} = 	an t$
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> **Subtangent** $= \dfrac{y}{dy/dx} = \dfrac{y}{	an t} = y\cot t$
> …

Q.Find the lengths of subtangent, subnormal at a point ' $t$ ' on the curve $x = a(Cos\,t + t\,Sin\,t)$ , $y = a(Sin\,t - t\,Cos\,t)$ .

Concept understanding — Tangents and Normals (Lengths)

Tangents and Normals to a Curve

Q.Find the lengths of subtangent, subnormal at a point 'ttt' on the curve x=a(Cos t+t Sin t)x = a(Cos\,t + t\,Sin\,t)x=a(Cost+tSint), y=a(Sin t−t Cos t)y = a(Sin\,t - t\,Cos\,t)y=a(Sint−tCost).

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Q.Find the lengths of subtangent, subnormal at a point ' $t$ ' on the curve $x = a(Cos\,t + t\,Sin\,t)$ , $y = a(Sin\,t - t\,Cos\,t)$ .

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