Let (x, y) be any point on the parabola y^2 = | vedclass

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(C) Let the coordinates of $P$,whose locus is to be determined,be $(h, k)$.Since $P$ divides the line segment joining $(0, 0)$ and $(x, y)$ in the rat

Vedclass · Accepted Answer

$y^2 = x$

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(C) Let the coordinates of $P$,whose locus is to be determined,be $(h, k)$.Since $P$ divides the line segment joining $(0, 0)$ and $(x, y)$ in the ratio $1 : 3$,by the section formula,the coordinates of $P$ are given by:$h = \frac{1 \cdot x + 3 \cdot 0}{1 + 3} = \frac{x}{4} \implies x = 4h$$k = \frac{1 \cdot y + 3 \cdot 0}{1 + 3} = \frac{y}{4} \implies y = 4k$Given that $(x, y)$ lies on the parabola $y^2 = 4x$,we substitute the expressions for $x$ and $y$:$(4k)^2 = 4(4h)$$16k^2 = 16h$$k^2 = h$Replacing $(h, k)$ with $(x, y)$,the locus of $P$ is $y^2 = x$.

List-$I$	List-$II$
$I$. Vertex	$A$. $8$
$II$. Length of latus rectum	$B$. $(\frac{29}{10}, \frac{-38}{10})$
$III$. Directrix	$C$. $3x+4y-1=0$
$IV$. One end of the latus rectum	$D$. $(\frac{-2}{5}, \frac{-16}{5})$
	$E$. $6$

Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be a point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$. Find the locus of $P$.

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