Understanding the Parabola and its Latus Rectum

The problem asks for the length of the latus rectum of a parabola, given information about its vertex and focus. To solve this, we need to recall the fundamental definitions and properties of a parabola.

A parabola is a set of all points in a plane that are equidistant from a fixed point (the focus, F) and a fixed line (the directrix).

• The vertex (V) is the point on the parabola that lies exactly midway between the focus and the directrix. It is the closest point on the parabola to both the focus and the directrix.
• The axis of symmetry is the line that passes through the vertex and the focus, and it is perpendicular to the directrix.
• The latus rectum is a special chord of the parabola that passes through the focus, is perpendicular to the axis of symmetry, and has its endpoints on the parabola.

The most crucial formula for this problem is the length of the latus rectum. For any parabola, the length of its latus rectum, denoted by $L$, is given by: $L = 4a$ where $a$ represents the distance between the vertex and the focus. This distance '$a$' is also known as the focal length. Our primary goal is to determine this value of '$a$' from the given information.

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Analyzing the Given Information and Visualizing the Parabola

Let's break down the information provided in the question:

1. "Vertex and focus are on the positive x-axis": This immediately tells us that both points lie on the x-axis, meaning their y-coordinates are 0. Since they are on the positive x-axis, their x-coordinates must be positive values. This implies the axis of symmetry of the parabola is the x-axis.

2. "Vertex (V) ... at a distance R ... from the origin": The origin is $(0,0)$. Since V is on the positive x-axis at a distance $R$, its coordinates must be $V = (R, 0)$.

3. "Focus (F) ... at a distance S (> R) respectively from the origin": Similarly, F is on the positive x-axis at a distance $S$. Thus, its coordinates are $F = (S, 0)$.

4. "Condition: S > R": This is a critical piece of information. Since both V and F are on the positive x-axis, and $S > R$, it means the focus F is located to the right of the vertex V.

Visualizing the Parabola: Given that the vertex is at $(R, 0)$ and the focus is at $(S, 0)$ with $S > R$, the parabola opens towards the positive x-axis (to the right). This mental image helps confirm our coordinate assignments and the direction of the focal length 'a'.

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Step-by-Step Solution

Step 1: Determine the Coordinates of the Vertex and Focus.

• Purpose: To calculate the distance '$a$' between the vertex and the focus, we first need to establish their exact positions in the Cartesian coordinate system. The origin $(0,0)$ serves as our reference point for the given distances.
• Vertex (V): The problem states the vertex is on the positive x-axis at a distance $R$ from the origin. Therefore, the coordinates of the vertex are: $V = (R, 0)$
• Focus (F): Similarly, the focus is on the positive x-axis at a distance $S$ from the origin. Therefore, the coordinates of the focus are: $F = (S, 0)$

Step 2: Calculate the Distance 'a' between the Vertex and the Focus.

• Purpose: The parameter '$a$' in the latus rectum formula $L = 4a$ is defined as the distance between the vertex and the focus. We will use the coordinates determined in Step 1 to calculate this distance.
• Method: Since both points $V(R, 0)$ and $F(S, 0)$ lie on the x-axis (their y-coordinates are the same), the distance between them is simply the absolute difference of their x-coordinates. $a = \sqrt{(S - R)^2 + (0 - 0)^2}$ $a = \sqrt{(S - R)^2}$
• Simplifying with absolute value: When taking the square root of a squared term, it's essential to use the absolute value to ensure the result is non-negative, as distance must always be positive: $a = |S - R|$
• Using the given condition ($S > R$): The problem explicitly states that $S > R$. This means that the quantity $(S - R)$ is a positive value. Therefore, we can remove the absolute value sign: $a = S - R$ This is the distance between the vertex and the focus, which is our focal length '$a$'.

Step 3: Apply the Latus Rectum Formula.

• Purpose: Now that we have successfully determined the value of '$a$', we can directly substitute it into the standard formula for the length of the latus rectum to find our answer.
• Formula: The length of the latus rectum $L$ is given by: $L = 4a$
• Substitution: Substitute the value of $a = (S - R)$ into the formula: $L = 4(S - R)$

Thus, the length of the latus rectum of the given parabola is $4(S - R)$.

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Tips and Common Mistakes to Avoid

1. Visualize the Parabola: Always begin by sketching or mentally visualizing the given information. Knowing that V and F are on the positive x-axis and $S > R$ immediately tells you the parabola opens to the right. This helps in correctly interpreting distances and directions, and avoids sign errors.
2. Definition of 'a': Remember that '$a$' is specifically the distance between the vertex and the focus. It must always be a positive value. If you were to calculate $R - S$ when $S > R$, you would get a negative value, which is incorrect for a distance. Always ensure you subtract the smaller coordinate from the larger one, or use the absolute value, i.e., $|S - R|$.
3. Latus Rectum Formula: Do not confuse the latus rectum length ($4a$) with other parabolic properties. Ensure you use the correct formula.
4. Coordinate System Placement: Be comfortable placing points on the coordinate axes based on their distances from the origin. If a point is on the positive x-axis at distance $D$, its coordinates are $(D, 0)$. If it were on the negative x-axis, it would be $(-D, 0)$.

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Summary and Key Takeaway

This problem is a straightforward application of the definition and properties of a parabola. The core idea is to correctly identify the focal length '$a$' (the distance between the vertex and the focus) using the given information about their positions relative to the origin. By understanding that the vertex and focus are on the positive x-axis at distances $R$ and $S$ respectively, and that $S > R$, we deduced that $a = S - R$. Once '$a$' is determined, the length of the latus rectum is simply $4a$.

The final answer is $\boxed{\text{4(S - R)}}$.