Question

A curve is such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2). The equation of the curve is –

• A. xy = 1
• B. xy = 2
• C. xy = 3
• D. None of these

Answer 1

Answer: (B)

xy = 2

Answer 2

Let P (x, y) be any point on the curve, PM the perpendicular to x-axis, PT the tangent at P meeting the axis of x at T. As given OT = 2. OM = 2x. Equation of the tangent at P (x, y) is

Y – y = $\frac{dy}{dx}$ (X – x)

It intersects the axis of x, where Y = 0

Ž – y = $\frac{dy}{dx}$ (X – x) Ž X = x – y $\frac{dx}{dy}$ = OT.

Hence, x – y $\frac{dx}{dy}$ = 2x Ž $\frac{dx}{x}$ + $\frac{dy}{y}$ = 0

Integrating, log x + log y = log c Ž xy = c.

This passes through (1, 2), \ c = 2.

\ The required curve is x y = 2.

Hence (2) is the correct answer.