For the curve xy = c^2,the subnormal at any p | vedclass

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(D) Given the curve equation: $xy = c^2$  $(i)$The formula for the subnormal at any point $(x, y)$ on a curve is given by: $	ext{Subnormal} = y \frac

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$y^3$

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(D) Given the curve equation: $xy = c^2$  $(i)$The formula for the subnormal at any point $(x, y)$ on a curve is given by: $	ext{Subnormal} = y \frac{dy}{dx}$From equation $(i)$,we have $y = \frac{c^2}{x}$.Differentiating with respect to $x$:$\frac{dy}{dx} = -\frac{c^2}{x^2}$Now,substitute $\frac{dy}{dx}$ into the subnormal formula:$	ext{Subnormal} = y \left( -\frac{c^2}{x^2} ight)$Since $x = \frac{c^2}{y}$,substitute $x$ in the expression:$	ext{Subnormal} = y \left( -\frac{c^2}{(\frac{c^2}{y})^2} ight) = y \left( -\frac{c^2}{\frac{c^4}{y^2}} ight)$$= y \left( -\frac{c^2 y^2}{c^4} ight) = -\frac{y^3}{c^2}$Since $c$ is a constant,the magnitude of the subnormal varies as $y^3$.

For the curve $xy = c^2$,the subnormal at any point varies as

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