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

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is

$y^{2}=4x \quad \text{(parabola)}$

$P \text{ divides } (0,0)\to(x,y) \text{ in } 1:3$

$\text{To find: locus of } P$

$y^{2}=4x:\ \text{vertex }(0,0),\ a=1,\ \text{opens right}$

$\text{e.g. } (x,y)=(1,2) \text{ satisfies } 2^{2}=4(1)$

$1.\ \text{section formula for } P$

$2.\ \text{write } x,y \text{ in terms of } P=(h,k)$

$3.\ \text{substitute in } y^{2}=4x \Rightarrow \text{locus}$

$P = \left(\dfrac{1\cdot x + 3\cdot 0}{1+3},\ \dfrac{1\cdot y + 3\cdot 0}{1+3}\right)$

$P = \left(\dfrac{x}{4},\ \dfrac{y}{4}\right)$

$P=(h,k):\quad h=\dfrac{x}{4},\quad k=\dfrac{y}{4}$

$\Rightarrow\ x=4h,\qquad y=4k$

$y^{2}=4x \ \Rightarrow\ (4k)^{2}=4(4h)$

$16k^{2}=16h \ \Rightarrow\ k^{2}=h$

$\text{locus: } y^{2}=x$

$\text{Answer: option (3)}$