Question

The equation of the curve which is such that the portion of the x-axis cut between the origin and tangent at any point is proportional to the ordinate of that point is (where $b$ is a constant of proportionality)

• A. $x=y(a-b\log y)$
• B. $\log x=b{y}^2+a$
• C. $x^2=y(a-b\log y)$
• D. $2\log x=by+a$

Answer 1

The correct option is A $x=y(a-b\log y)$

Let the equation of the curve be $y=f\left(x\right)$.

It is given that $OT\propto y$
or $OT=by$
or $OM-TM=by$
or $x-\frac{y}{\frac{dy}{dx}}=by$ [$\because$ TM = Length of the sub-tangent]
or $x-y\frac{dx}{dy}=by$
or $\frac{dx}{dy}-\frac{x}{y}=-b$
It is linear differential equation.
Its solution is $\frac{x}{y}=-b\log y+a$
or $x=y(a-b\log y)$