Understanding the Problem

We are given a parabola with equation $x^2=8y$. Let O be the vertex (which is at (0,0)) and Q be any point on this parabola. Point P divides the line segment OQ internally in the ratio 1:3. We need to find the locus of point P.

Step-by-Step Solution

Step 1: Parameterize Point Q on the Parabola

The standard form of a parabola is $x^2=4ay$. Comparing this with the given equation $x^2=8y$, we find that $4a=8$, so $a=2$.

A standard parametric form for a point Q on this parabola is: $Q=(2at,a{t}^2)=(4t,2t^2)$, where $t$ is a parameter.

Therefore, the coordinates of Q are $(4t,2t^2)$.

Step 2: Coordinates of Vertex O

The vertex O of the parabola is at the origin: $O=(0,0)$.

Step 3: Find Coordinates of Point P using Section Formula

Point P divides the line segment joining O(0,0) and Q(4t, 2t²) internally in the ratio 1:3.

Using the section formula for internal division (m:n = 1:3), the coordinates of P(x, y) are given by:

$x=\frac{m{x}_2+n{x}_1}{m+n}=\frac{1\times (4t)+3\times (0)}{1+3}=\frac{4t}{4}=t$

$y=\frac{m{y}_2+n{y}_1}{m+n}=\frac{1\times (2t^2)+3\times (0)}{1+3}=\frac{2t^2}{4}=\frac{t^2}{2}$

So, the coordinates of P are $(x,y)=(t,\frac{t^2}{2})$.

Step 4: Eliminate the Parameter 't' to Find the Locus

From Step 3, we have two equations:

1. $x=t$

2. $y=\frac{t^2}{2}$

Substitute Equation 1 into Equation 2 to eliminate the parameter $t$:

$y=\frac{x^2}{2}$

We can rewrite this equation as:

$x^2=2y$

Step 5: Final Answer

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

Comparing with the given options, the correct answer is: x² = 2y

Related Topics and Formulae

Key Concepts

• Parabola: A conic section defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).
• Standard Equation: The standard form of a parabola with vertex at the origin and vertical axis is $x^2=4ay$, where 'a' is the distance from the vertex to the focus.
• Parametric Form: For the parabola $x^2=4ay$, a point can be represented as $(2at,a{t}^2)$.
• Section Formula: Used to find the coordinates of a point dividing a line segment internally in a given ratio m:n. If a point P divides the line joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then the coordinates of P are: $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$.
• Locus: The path traced by a moving point under a given condition. Finding the locus often involves eliminating the parameter from the parametric equations.