The length of the normal to the curve `Y=a((e^(-x//a)+e^(x//a))/(2))` at any point varies as the :

Options

A. abscissa of the point
B. ordinate of the point
C. square of the abscissa of the point
D. square of the ordinate of the point

Answer

D

Explanation

We have,
`y=a((e^(-x//a)+e^(x//a))/(2))rArr (dy)/(dx)=(e^(x//a)-e^(-x//a))/(2)`
Length of the normal at any point `=y sqrt(1+((dy)/(dx))^(2))`
`=y sqrt(1+((e^(x//a)-e^(-x//a))/(2))^(2))=y((e^(x//a)+e^(-x//a))/(2))=y((2y)/(a))=(2y^(2))/(a)`
Hence, the length of the normal at any point varies as the square of the ordinate of the point.