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(C) The equation of a parabola with the origin as its focus and the $X$-axis as its axis is given by $y^2 = 4a(x+a)$,where $a$ is a parameter. Differe

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$3$

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(C) The equation of a parabola with the origin as its focus and the $X$-axis as its axis is given by $y^2 = 4a(x+a)$,where $a$ is a parameter. Differentiating with respect to $x$,we get $2y \frac{dy}{dx} = 4a$,which implies $a = \frac{y}{2} \frac{dy}{dx}$. Substituting the value of $a$ back into the original equation: $y^2 = 4 \left( \frac{y}{2} \frac{dy}{dx} \right) \left( x + \frac{y}{2} \frac{dy}{dx} \right)$ $y^2 = 2y \frac{dy}{dx} \left( x + \frac{y}{2} \frac{dy}{dx} \right)$ $y^2 = 2xy \frac{dy}{dx} + y^2 \left( \frac{dy}{dx} \right)^2$. Here,the highest order derivative is $\frac{dy}{dx}$,so the order $m = 1$. The power of the highest order derivative is $2$,so the degree $n = 2$. Therefore,$mn - m + n = (1)(2) - 1 + 2 = 2 - 1 + 2 = 3$.

Column-$I$	Column-$II$
$(A)$ Blue-green algae	$(i)$ Spirogyra
$(B)$ Unicellular algae	$(ii)$ Nostoc
$(C)$ Filamentous algae	$(iii)$ Volvox
$(D)$ Colonial algae	$(iv)$ Chlamydomonas

If $m$ and $n$ are respectively the order and degree of the differential equation of the family of parabolas with origin as its focus and $X$-axis as its axis,then $mn-m+n=$

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