If a curve passes through the point (1,1) and | vedclass

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(D) Let the equation of the curve be $y=f(x)$.Since the curve passes through the point $(1,1)$,we have $f(1)=1$.According to the problem,at any point

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$(2,2)$

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(D) Let the equation of the curve be $y=f(x)$.Since the curve passes through the point $(1,1)$,we have $f(1)=1$.According to the problem,at any point $(x, y)$ on the curve,the product of the slope of its tangent $(\frac{dy}{dx})$ and the $x$-coordinate is equal to the $y$-coordinate.Thus,$x \cdot \frac{dy}{dx} = y$.Rearranging the terms,we get $\frac{dy}{y} = \frac{dx}{x}$.Integrating both sides,we obtain $\int \frac{dy}{y} = \int \frac{dx}{x}$,which gives $\ln|y| = \ln|x| + C$.This simplifies to $y = kx$,where $k = e^C$.Using the condition $f(1)=1$,we substitute $x=1$ and $y=1$ into $y=kx$ to get $1 = k(1)$,so $k=1$.The equation of the curve is $y=x$.Checking the given options,the point $(2,2)$ satisfies the equation $y=x$.

If a curve passes through the point $(1,1)$ and at any point $(x, y)$ on the curve,the product of the slope of its tangent and $x$ coordinate of the point is equal to the $y$ coordinate of the point,then the curve also passes through the point

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