Let O be the vertex and Q be any point on the | vedclass

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(A) The vertex $O$ of the parabola $x^2 = 8y$ is $(0, 0)$.Let the coordinates of point $Q$ on the parabola be $(4t, 2t^2)$,as it satisfies $x^2 = 8y$

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$x^2 = 2y$

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(A) The vertex $O$ of the parabola $x^2 = 8y$ is $(0, 0)$.Let the coordinates of point $Q$ on the parabola be $(4t, 2t^2)$,as it satisfies $x^2 = 8y$ (i.e.,$(4t)^2 = 16t^2 = 8(2t^2)$).Let $P(h, k)$ be the point that divides the line segment $OQ$ internally in the ratio $1:3$.Using the section formula,the coordinates of $P$ are:$h = \frac{1 	imes 4t + 3 	imes 0}{1 + 3} = \frac{4t}{4} = t$$k = \frac{1 	imes 2t^2 + 3 	imes 0}{1 + 3} = \frac{2t^2}{4} = \frac{t^2}{2}$From $h = t$,we have $t = h$.Substituting this into the expression for $k$:$k = \frac{h^2}{2} \implies h^2 = 2k$.Replacing $(h, k)$ with $(x, y)$,the locus of $P$ is $x^2 = 2y$.

Let $O$ be the vertex and $Q$ be any point on the parabola $x^2 = 8y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$,then the locus of $P$ is:

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