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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).
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Solution
Portion of the x-axis cut off between the origin and tangent at a point \[= x - y \hspace{0.167em} \hspace{0.167em} \frac{dx}{dy} = OT\]
It is given, OT = 2x
\[\begin{array}{l}\therefore \hspace{0.167em} \hspace{0.167em} x - y \hspace{0.167em} \hspace{0.167em} \frac{dx}{dy} = 2x \\ - x = y\frac{dx}{dy} \\ - \int\frac{dx}{x} = \int\frac{dy}{y} \\ \therefore \hspace{0.167em} \hspace{0.167em} xy = k\end{array}\]
Since the curve passes through the point (1, 2)
⇒ at x = 1 ⇒ y = 2
∴ k = 2
∴ xy = 2
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