Question

The length of the normal to the curve $y=a\cosh \left(\frac{x}{a}\right)$ at any point varies as

• A. Ordinate
• B. Abscissa
• C. Square of the abscissa
• D. Square of the ordinate

Answer 1

Answer: (D)

Square of the ordinate

We know,
$\frac{\delta y}{\delta x}=\frac{\delta a}{\delta x}\cosh \frac{x}{a}$
$=a.\sinh \left(\frac{x}{a}\right).\left(\frac{1}{a}\right)$
$=\sinh \left(\frac{x}{a}\right)$
Now length of normal $PN=\left|y\right|\sqrt{1+{\left(\frac{\delta y}{\delta x}\right)}^2}$
$PN=y\sqrt{1+{\sinh }^2\left(\frac{x}{a}\right)}(\because {\cosh }^{2}x=1+{\sinh }^{}{\sinh }^{2}x)$
$=y\sqrt{{\cosh }^2\frac{x}{a}}=y\cosh \frac{x}{a}$
$PN=\frac{y^2}{a}(\cosh \frac{x}{a}=\frac{y}{a})$
$\Rightarrow PN\propto y^2$