Question

Find the equation of the curve 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).

Answer 1

Portion of the x-axis cut off between the origin and tangent at a point $=x-y\hspace{0.17em}\hspace{0.17em}\frac{dx}{dy}=\mathrm{OT}$

It is given, OT = 2x

$\begin{array}{l}\therefore \hspace{0.17em}\hspace{0.17em}x-y\hspace{0.17em}\hspace{0.17em}\frac{dx}{dy}=\mathrm{2}x\\ -x=y\frac{dx}{dy}\\ -\int \frac{dx}{x}=\int \frac{dy}{y}\\ \therefore \hspace{0.17em}\hspace{0.17em}xy=k\end{array}$

Since the curve passes through the point (1, 2)
⇒ at x = 1 ⇒ y = 2
∴ k = 2
∴ xy = 2