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  4. If at any point on a curve the subtangent and subnormal are equal, then the tangent is equal to:

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Q. If at any point on a curve the subtangent and subnormal are equal, then the tangent is equal to:

Application of Derivatives

Solution:

we have, subtangent = subnormal
$\Rightarrow \frac{y}{\frac{dy}{dx}} =y. \frac{dy}{dx}$
$ \Rightarrow \left(\frac{dy}{dx}\right)^{2} = 1 \Rightarrow \frac{dy}{dx} = \pm1 $ .....(1)
So, length of the tangent $= \frac{y\sqrt{1+ \left(\frac{dy}{dx}\right)^{2}}}{\frac{dy}{dx}} $
$\Rightarrow $ length of the tangent = $ \sqrt{2} y$
[using equation (1)]
$ \Rightarrow $ length of the tangent = $\sqrt{2}$ ordinate