The differential equation of all parabolas wi | vedclass

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(D) The general equation of a parabola with its axis parallel to the $y$-axis is given by $(x-h)^{2} = 4a(y-k)$,where $h, k,$ and $a$ are arbitrary co

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None of these

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(D) The general equation of a parabola with its axis parallel to the $y$-axis is given by $(x-h)^{2} = 4a(y-k)$,where $h, k,$ and $a$ are arbitrary constants. Since there are $3$ arbitrary constants,we need to differentiate the equation $3$ times. Let $(x-h)^{2} = A(y-k)$,where $A = 4a$. Differentiating with respect to $x$: $2(x-h) = A y_{1}$. Differentiating again with respect to $x$: $2 = A y_{2}$. Differentiating a third time with respect to $x$: $0 = A y_{3}$. Since $A = 4a 
eq 0$,we must have $y_{3} = 0$. None of the given options match $y_{3} = 0$.

The differential equation of all parabolas with axis parallel to the $y$-axis is

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