For a curve,what is the value of frac{(text{L | vedclass

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(A) Let the curve be $y = f(x)$.At any point $(x, y)$ on the curve,let $m = \frac{dy}{dx}$ be the slope of the tangent.The length of the tangent is gi

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$\frac{	ext{Subnormal}}{	ext{Subtangent}}$

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(A) Let the curve be $y = f(x)$.At any point $(x, y)$ on the curve,let $m = \frac{dy}{dx}$ be the slope of the tangent.The length of the tangent is given by $L_T = |y| \sqrt{1 + \frac{1}{m^2}} = |y| \frac{\sqrt{1 + m^2}}{|m|}$.The length of the normal is given by $L_N = |y| \sqrt{1 + m^2}$.The length of the subtangent is $S_T = |\frac{y}{m}|$.The length of the subnormal is $S_N = |ym|$.Now,consider the ratio $\frac{(L_N)^2}{(L_T)^2} = \frac{y^2(1 + m^2)}{y^2 \frac{1 + m^2}{m^2}} = m^2$.Also,the ratio of subnormal to subtangent is $\frac{S_N}{S_T} = \frac{|ym|}{|y/m|} = |m^2| = m^2$.Therefore,$\frac{(L_N)^2}{(L_T)^2} = \frac{S_N}{S_T}$.

For a curve,what is the value of $\frac{(\text{Length of Normal})^2}{(\text{Length of Tangent})^2}$?

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