Find the length of subnormal at x= 2 on the curve y = $x^{3}$ .
96

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Solution

The correct option is A 96

We saw that P₁N is the length of subnormal in the given figure. It It can be easily calculated from the triangle PNP₁.

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

$θ$ is the slope of the tangent

From the figure tan( $θ = \frac{P_{1} N}{P P_{1}}$ )
=> P₁N = PP₁ tan $θ$

To calculate tan $θ$ , we will differentiate y

=> f’(x) = 3 $x^{2}$

=> f’(2) = 12 = tan $θ$

PP₁ = y coordinate = $2^{3}$ = 8
=> Length of subnormal = PP₁ tan $θ$ = 8 x 12 = 96

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