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Question
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
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Solution
According to the question,
\[y\frac{dy}{dx} = x\]
\[ \Rightarrow y dy = x dx\]
Integrating both sides with respect to x, we get
\[\int y dy = \int x dx\]
\[ \Rightarrow \frac{y^2}{2} = \frac{x^2}{2} + C\]
\[\text{ Since the curve passes through }\left( 0, a \right),\text{ it satisfies the above equation . }\]
\[ \therefore \frac{a^2}{2} = \frac{0}{2} + C\]
\[ \Rightarrow C = \frac{a^2}{2}\]
Putting the value of C, we get
\[\frac{y^2}{2} = \frac{x^2}{2} + \frac{a^2}{2}\]
\[ \Rightarrow x^2 - y^2 = - a^2\]
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