The equation of the family of curves for whic | vedclass

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(B) The length of the subnormal at any point $(x, y)$ is given by $|y \frac{dy}{dx}| = k$. This implies $y \frac{dy}{dx} = \pm k$. Separating the vari

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$y^2 - A = 2Kx$

Vedclass · Answer

(B) The length of the subnormal at any point $(x, y)$ is given by $|y \frac{dy}{dx}| = k$. This implies $y \frac{dy}{dx} = \pm k$. Separating the variables,we get $y \, dy = \pm k \, dx$. Integrating both sides,we obtain $\int y \, dy = \int \pm k \, dx$. This results in $\frac{y^2}{2} = \pm kx + C$,where $C$ is the constant of integration. Multiplying by $2$,we get $y^2 = \pm 2kx + 2C$. Letting $A = 2C$,we have $y^2 - A = \pm 2kx$. Thus,the equation of the family of curves is $y^2 - A = 2Kx$ (considering the positive constant form).

The equation of the family of curves for which the length of the subnormal at any point $(x, y)$ is always a constant $(k)$ is

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