A curve having the condition that the slope o | vedclass

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(C) Let the point on the curve be $(x, y)$.The slope of the tangent at $(x, y)$ is given by $\frac{dy}{dx}$.The slope of the line joining $(x, y)$ to

Vedclass · Accepted Answer

Parabola

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(C) Let the point on the curve be $(x, y)$.The slope of the tangent at $(x, y)$ is given by $\frac{dy}{dx}$.The slope of the line joining $(x, y)$ to the origin $(0, 0)$ is $\frac{y - 0}{x - 0} = \frac{y}{x}$.According to the given condition,$\frac{dy}{dx} = 2 \left( \frac{y}{x} \right)$.This is a variable separable differential equation: $\frac{dy}{y} = 2 \frac{dx}{x}$.Integrating both sides,we get $\int \frac{dy}{y} = 2 \int \frac{dx}{x}$.$\ln|y| = 2 \ln|x| + C$,which can be written as $\ln|y| = \ln|x^2| + \ln|c|$.Thus,$y = cx^2$,which represents a parabola.

$A$ curve having the condition that the slope of the tangent at any point $(x, y)$ is two times the slope of the straight line joining the same point to the origin $(0, 0)$ is a/an:

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