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 \(S\) (\(S>R\)) respectively from the origin, is :

• A. \(2(S + R)\)
• B. \(2(S - R)\)
• C. \(4(S + R)\)
• D. \(4(S - R)\)

Hint:

The focus-vertex distance is the fundamental parameter of a parabola. Once you have \(a\), the latus rectum is always \(4a\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
For a parabola, the distance between the vertex (\(V\)) and the focus (\(F\)) is denoted by \(a\). The length of the latus rectum is \(4a\).
Step 2: Detailed Explanation:
1. Let the origin be \(O(0, 0)\).
2. The vertex \(V\) is on the positive \(x\)-axis at distance \(R\), so \(V = (R, 0)\).
3. The focus \(F\) is on the positive \(x\)-axis at distance \(S\), so \(F = (S, 0)\).
4. The distance \(a = VF = |S - R|\). Since \(S>R\), \(a = S - R\).
5. The length of the latus rectum is given by \(4a\).
Substituting the value of \(a\):
\[ \text{Length of L.R.} = 4(S - R) \]
Step 3: Final Answer:
The length is \(4(S - R)\), which is option (D).