Understanding the Problem

We are given a parabola with the equation $y^2=4x$. Let $A=(0,0)$ be the origin and $B=(x,y)$ be any point on this parabola. A point $P$ divides the line segment $AB$ internally in the ratio 1:3. We need to find the locus of point $P$.

Step-by-Step Solution

Step 1: Find the Coordinates of Point P

Point $P$ divides the segment joining $A(0,0)$ and $B(x,y)$ in the ratio m:n = 1:3.

The coordinates of such a point are given by the section formula:

$P=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})=(\frac{1\cdot x+3\cdot 0}{1+3},\frac{1\cdot y+3\cdot 0}{1+3})$

Simplifying the coordinates:

$P=(\frac{x}{4},\frac{y}{4})$

Let the coordinates of $P$ be $(h,k)$. Therefore:

$h=\frac{x}{4}$ and $k=\frac{y}{4}$

Step 2: Express x and y in Terms of h and k

From the equations above, we can solve for the original coordinates $x$ and $y$ of the parabola:

$x=4h$

$y=4k$

Step 3: Substitute into the Parabola's Equation

We know that point $B(x,y)$ lies on the parabola $y^2=4x$. Let's substitute the expressions for $x$ and $y$ from Step 2 into this equation.

$y^2=4x$

$(4k{)}^2=4\cdot (4h)$

$16k^2=16h$

Step 4: Simplify the Equation

We can divide both sides of the equation by 16 to simplify it.

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

$k^2=h$

Step 5: Write the Locus

The equation $k^2=h$ represents the relationship between the coordinates of point $P(h,k)$. To express this as a standard locus, we replace $h$ with $x$ and $k$ with $y$.

Therefore, the locus of point $P$ is:

$y^2=x$

Final Answer

The locus of point P is $y^2=x$.

This corresponds to the option: y² = x

Related Topics & Formulae

1. Section Formula

The coordinates of the point $P(x,y)$ that divides the line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$ are given by: