The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a

Q: Wait, why is the distance $a$ equal to $S - R$? Shouldn't it be $S + R$?

Think of the x-axis as a number line. If you are at position $R$ and you want to reach position $S$, and $S$ is further to the right, the distance you cover is the difference between the two coordinates: $S - R$. Adding them would only make sense if they were on opposite sides of the origin, but here both are on the positive x-axis.

Q: Does the position of the vertex relative to the origin change the length of the latus rectum?

That is a fantastic question! The length of the latus rectum is an intrinsic property of the parabola's shape, determined solely by the distance between the vertex and the focus ($a$). Shifting the parabola along the x-axis (changing $R$) changes its location, but it does not stretch or compress the curve itself. So, the length remains $4a$ regardless of $R$.

Q: Why does the parabola open to the right?

The axis of symmetry is the x-axis. The focus is always 'inside' the bowl of the parabola. Since the focus $F(S, 0)$ is to the right of the vertex $V(R, 0)$, the parabola must curve around the focus, forcing it to open towards the positive x-direction.

Q: Is the latus rectum always $4a$?

Yes, for any parabola defined by the standard equation $(y - k)^2 = 4a(x - h)$, the length of the chord passing through the focus perpendicular to the axis of symmetry is always $4a$. This is a fundamental geometric property derived from the definition of a parabola as a locus of points.

Q: What if $S < R$? Would the length be negative?

Length must always be a positive scalar. If $S R$, we use $S - R$ to ensure the length is positive.

Question

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S(> R)$ respectively from the origin, is:

Options:

(A) $4(S + R)$

(B) $2(S - R)$

(C) $4(S - R)$

(D) $2(S + R)$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the $x$-axis. You walk a distance $R$ from the origin and place a marker at the vertex, $V(R, 0)$. You then continue walking along the positive $x$-axis until you reach a distance $S$, where you place the focus, $F(S, 0)$.

Because we are given $S > R$, the focus sits to the right of the vertex. In the geometry of conic sections, the parabola must always wrap itself around the focus. Therefore, our parabola opens to the right.

### Defining the Parameter $a$

In the study of conic sections, the parameter $a$ is the bridge between the geometry of the curve and its algebraic equation. It represents the focal length, which is the distance between the vertex and the focus.

Mathematically, we calculate this distance by finding the difference between the $x$-coordinates of our two points:

$$a = S - R$$

Think of this as the "scale" of your parabola. If $a$ is large, the parabola is wide and flat; if $a$ is small, the parabola is tight and sharp. By identifying $a = S - R$, we have successfully translated the physical description into the language of mathematics.

### The Elegance of the Latus Rectum

We now consider the latus rectum. By definition, the latus rectum is the chord that passes through the focus, perpendicular to the axis of symmetry, with its endpoints resting on the parabola. It represents the "width" of the parabola at its focal point.

For any standard parabola, the geometry dictates that this length is exactly four times the focal length. This arises from the definition of the parabola as the locus of points equidistant from the focus and the directrix. The derivation yields the following result:

$$LLR = 4a$$

### Final Calculation

We are now ready to synthesize our findings. We have the focal length $a = S - R$ and the formula for the length of the latus rectum $LLR = 4a$.

By substituting our expression for $a$ into the formula, we obtain the final result:

$$LLR = 4(S - R)$$

This result is simple, clean, and perfectly logical. It confirms that the width of the parabola is directly proportional to the gap between the vertex and the focus. By mapping the vertex and focus, we have successfully uncovered the anatomy of the parabola.

Animated Solution for Mathematics - Conic Sections: The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$ -axis at a distance $R$ and $S (> R)$ respectively from the origin, is:

Select Answer:

Visualized Solution

Coordinate Axes Setup

Plotting the Vertex $V$

Plotting the Focus $F$

Relative Position of $V$ and $F$

Orientation of the Parabola

The Parameter $a$

Calculating Distance $a$

The Latus Rectum

Length of Latus Rectum ( $LL R$ )

Substituting $a$

Final Answer

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

Analyzing the Setup

Defining the Parameter $a$

The Elegance of the Latus Rectum

Final Calculation

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Select Answer:

Visualized Solution

Coordinate Axes Setup

Plotting the Vertex .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } V

Plotting the Focus .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } F

Relative Position of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } V and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } F