Question

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

• A. $y = \frac{x}{(a - b\log x)}$
• B. $\log x = by^{2} + a$
• C. $x^{2} = y(a - b\log y)$
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Tangent at P(x, y) to the curve y = f(x) may be expressed as $Y - y = \frac{dy}{dx}(X - x)$

∴ $Q = \left( x - y\frac{dx}{dy},0 \right)$

As per question, $OQ \propto y$

⇒ $x - y\frac{dx}{dy} \propto y$ ⇒ $x - y\frac{dx}{dy} = by$ ⇒ $\frac{x}{y} - \frac{dx}{dy} = b$

∴ $\frac{dx}{dy} = \frac{x}{y} - b$

Let $\frac{x}{y} = v$ ⇒ x = vy ⇒ $\frac{dx}{dy} = v + y\frac{dv}{dy}$ ⇒ $\frac{x}{y} - b = v + y\frac{dv}{dy}$

⇒ $v - b = v + y\frac{dv}{dy}$ ⇒ $- b = y\frac{dv}{dy}$ ⇒ $- b\frac{dy}{y} = dv$

Integrating, $\int_{}^{}{dv = - b\int_{}^{}\frac{dy}{y}}$ ⇒ $v = - b\ln y + a$

⇒ $\frac{x}{y} = a - b\ln y$ (a, an arbitrary constant)

∴ $x = y(a - b\ln y)$