In the corve `y=ce^(x//a),` the

Options

A. subtangent is constant
B. subnormal varies as the square of the ordinate
C. tangent at `(x_(1),y_(1))` on the curve intersects the x-axis at a distance of ` (x_(1)-a) ` from the origin.
D. equation of normal at the point where the curve cuts y-axis is ` cy+ax=c `

Answer

D

Explanation

We have,
` y=ce^(x//a) rArr (dy)/(dx)=(c)/(a)e^(x//a) rArr (dy)/(dx)=(1)/(a)y `
` therefore (y)/(dy//dx)=a=" Const. " rArr " Subagent " = " Const. " `
Length of the subnormal
`= y(dy)/(dx) =y xx (y)/(a) =(y^(2))/(a) prop` (Square of the ordinate)
Equation of the tangent at ` (x_(1),y_(1)) ` is
` y-y_(1)=(y_(1))/(a)(x-x_(1)) `
This meets x- axis at a point given by
`-y=(y_(1))/(a)(x-x_(1)) rArr x=x_(1)-a. `
The curve meets y- axis at (0,c)
` therefore ((dy)/(dx))_((0","c))=c//a `
So, equation of the normal at (0,c) is
` y-c=-(1)/(c//a)(x-0) rArr ax+cy=c^(2) `