If at any point on a curve the subtangent and | vedclass

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Question

(A) Let $P(x_1, y_1)$ be a point on the curve $y = f(x)$.Let $m = \frac{dy}{dx}$ be the slope of the tangent at $P$.The length of the subtangent is gi

Vedclass · Accepted Answer

$\sqrt{2} 	imes 	ext{ordinate}$

Vedclass · Answer

(A) Let $P(x_1, y_1)$ be a point on the curve $y = f(x)$.Let $m = \frac{dy}{dx}$ be the slope of the tangent at $P$.The length of the subtangent is given by $\left| \frac{y_1}{m} ight|$ and the length of the subnormal is given by $|y_1 m|$.Given that the subtangent and subnormal are equal,we have:$\left| \frac{y_1}{m} ight| = |y_1 m|$$\Rightarrow \frac{1}{|m|} = |m|$$\Rightarrow m^2 = 1 \Rightarrow m = \pm 1$.The length of the normal at point $P$ is given by $|y_1| \sqrt{1 + m^2}$.Substituting $m^2 = 1$,we get:$	ext{Length of normal} = |y_1| \sqrt{1 + 1} = |y_1| \sqrt{2} = \sqrt{2} 	imes 	ext{ordinate}$.

If at any point on a curve the subtangent and subnormal are equal,then the length of the normal is equal to

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