The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point (1,1) is

Options

A. `ye^(x//y)=e`
B. `xe^(x//y)=e`
C. `xe^(y//x)=e`
D. `ye^(y//x)=e`

Answer

A

Explanation

Equation of tangent is `Y-y=(dy)/(dx)(X-x)`
For x-intercept, Y=0 . Thus, `X=x-y(dx)/(dy)`.
According to question, `x-y(dx)/(dy)=y`
or `(dy)/(dx)=y/(x-y)`
Putting `y=vx`, we get
`v+x(dv)/(dx)=v/(1-v)`
or `x(dv)/(dx)=v/(1-v)-v=(v-v+v^(2))/(1-v)`
or `int(1-v)/(v^(2))dv=int(dx)/x`
or `-x/y-logy/x=logx+c`
Given when `x=1, y=1`, then `c=-1`
Hence, equation of curve is `1-x/y=logy`
or `y=ee^(-x//y)`
or `ye^(x//y)=e`