The length of the normal to the curve $y=a\left(\frac{e^{-x / a}+e^{x / a}}{2}\right)$ at any point varies as the |
abscissa of the point ordinate of the point square of the abscissa of the point square of the ordinate of the point |
square of the ordinate of the point |
We have, $y=a\left(\frac{e^{-x / a}+e^{x / a}}{2}\right) \Rightarrow \frac{d y}{d x}=\frac{e^{x / a}-e^{-x / a}}{2}$ ∴ Length of the normal at any point = $y \sqrt{1+\left(\frac{d y}{d x}\right)^2}$ $=y \sqrt{1+\left(\frac{e^{x / a}-a^{-x / a}}{2}\right)^2}=y\left(\frac{e^{x / a}+e^{-x / a}}{2}\right)=y\left(\frac{2 y}{a}\right)=\frac{2 y^2}{a}$ Hence, the length of the normal at any point varies as the square of the ordinate of the point. |