Question

Find the length of subnormal at x= 2 on the curve y = $x^3$.

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Answer 1

We saw that P₁N is the length of subnormal in the given figure. It It can be easily calculated from the triangle P₁NP.

PP₁ is the y-coordinate of the point where the tangent is drawn.

$\theta$ is the slope of the tangent $\theta$ From the figure tan $\theta =\frac{P_{1}N}{P{P}_1}$ $\Rightarrow P_{1}N=P{P}_{1}tan\theta$ To calculate tan $\theta$, we will differentiate y $\Rightarrow$ f'(x) = 3 $x^2$ $\Rightarrow$ f'(2) = 12 = tan $\theta$ $P{P}_1=y$ coordinate = $2^3$ = 8 $\Rightarrow$ Length of subnormal = $P{P}_{1}tan\theta =8\times 12=96$