lf length of tangent at any point on th curve `y=f(x)` intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

Explanation

Given, length of tangent to curve `y=f(x)` is 1.
`therefore |ysqrt(1+((dy)/(dx))^(2))/((dy)/(dx))|=1`
or `(y^(2))1+((dy)/(dx))^(2)=((dy)/(dx))^(2)`
or `((dy)/(dx))^(2)=y^(2)/(1-y^(2))`
or `(dy)/(dx) =+-y/sqrt(1-y^(2))`
or `intsqrt(1-y^(2))/(y) dy=int+-dx`
Put `y=sintheta` so that `dy=costhetad(theta)`
`therefore int(costheta)/(sintheta) costheta d(theta) = +-x+c`
or `int("cosec "theta-sintheta)d(theta)=+-x+c`
or `log|"cosec "theta-cottheta|+costheta=+-x+c`
or `log|(1-sqrt(1-y^(2))/(y))|+sqrt(1-y^(2))=+-x+c`