Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment fr

Q: Why do we use the section formula instead of just dividing the equation by 4?

Great question! Dividing the equation by 4 would only scale the parabola, but the section formula gives us the locus of a point that is defined by a specific geometric relationship (the ratio $1:3$). It is about mapping the movement of $Q$ to the movement of $P$.

Q: Can I use parametric coordinates like $(t^2, 2t)$?

Absolutely! Using $Q(t^2, 2t)$ is a brilliant shortcut. If $P$ divides $OQ$ in $1:3$, then $P$ would be $(\frac{t^2}{4}, \frac{2t}{4}) = (\frac{t^2}{4}, \frac{t}{2})$. You can then eliminate $t$ to find the locus. Both methods lead to the same beautiful result.

Q: Why is the final locus $y^2 = x$ and not $y^2 = 4x$?

Because the point $P$ is closer to the origin than $Q$. Since $P$ is only $1/4$ of the distance, its coordinates are smaller, which means the parabola it traces must be 'tighter' or 'narrower' than the original one. The equation $y^2 = x$ reflects this change.

Question

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

Options:

(A) $x^2 = y$

(B) $y^2 = 2x$

(C) $y^2 = x$

(D) $x^2 = 2y$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow explorer of the mathematical universe! Today, we are not just solving a problem; we are embarking on a journey to understand how shapes transform in the coordinate plane.

Imagine you are standing at the origin of a coordinate system, looking at a beautiful, sweeping parabola defined by the equation $y^2 = 4x$. This parabola is our canvas, a fixed path where any point $Q(x, y)$ can exist.

We are looking for the path, or the locus, of a point $P(h, k)$ that is intimately connected to $Q$. Specifically, $P$ is the point that divides the segment connecting the origin $O(0, 0)$ to $Q(x, y)$ in a $1:3$ ratio.

### The Bridge: The Section Formula

To capture this relationship, we use the section formula. When a point $P$ divides a line segment joining $O(0, 0)$ and $Q(x, y)$ in a ratio $m:n$, its coordinates are given by the weighted average of the endpoints.

Here, our ratio is $1:3$, so $m=1$ and $n=3$. The coordinates of $P(h, k)$ are given by:

$$h = \frac{1 \cdot x + 3 \cdot 0}{1 + 3}, \quad k = \frac{1 \cdot y + 3 \cdot 0}{1 + 3}$$

This simplifies beautifully to $h = \frac{x}{4}$ and $k = \frac{y}{4}$. Think of this as a scaling transformation where every point on the original parabola is being 'shrunk' by a factor of 4 towards the origin.

### The Transformation: From $Q$ to $P$

Now, we have the coordinates of $P$ in terms of $Q$. To find the equation for $P$ itself, we express $x$ and $y$ in terms of $h$ and $k$:

$$x = 4h, \quad y = 4k$$

Since $Q(x, y)$ is a point on the parabola $y^2 = 4x$, it must satisfy that equation. By substituting our expressions for $x$ and $y$ into the parabola's equation, we obtain:

$$(4k)^2 = 4(4h)$$

### The Revelation: A New Parabola

Now, let us perform the final algebraic dance. Expanding the left side, we get $16k^2$, and on the right side, we have $4 \cdot 4h$, which is $16h$.

Our equation becomes:

$$16k^2 = 16h$$

Dividing both sides by 16, we arrive at the stunningly simple result: $k^2 = h$. To express this as a general locus, we replace $h$ and $k$ with the standard variables $x$ and $y$.

The final equation of the locus is **$y^2 = x$**. We have discovered that the locus of $P$ is another parabola, one that is narrower and closer to the origin than the original.

Animated Solution for Mathematics - Conic Sections: Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . 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

Select Answer:

Visualized Solution

Visualizing the Given Parabola

Defining Point $Q$ and Segment $O Q$

Locating the Point $P (h, k)$

The Section Formula

Applying the Section Formula

Simplifying the Coordinates

Expressing $x$ and $y$ via $h$ and $k$

Substituting into the Parabola Equation

Expanding the Equation

Simplifying the Locus Equation

The Final Locus

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Bridge

The Transformation

The Revelation

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