Question

In hyperbola, the length of the transverse axis is the distance between:

• A. two vertices
• B. two foci
• C. vertex and the origin
• D. focus and the vertex

Hint:

Remember these standard distance relationships for conics:
- Distance between vertices = Length of transverse/major axis (\(2a\)).
- Distance between foci = Focal distance (\(2ae\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A hyperbola is a conic section defined as the locus of points where the absolute difference in distances to two fixed points (foci) is constant.
It consists of two main axes of symmetry: the transverse axis (which passes through both foci and vertices) and the conjugate axis (which is perpendicular to it).

Step 3: Detailed Explanation:
Let us consider a standard hyperbola with the equation:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
The key features of this hyperbola are:
1. Center: Located at the origin \((0, 0)\).
2. Foci: Located at \(F_1(ae, 0)\) and \(F_2(-ae, 0)\), where \(e > 1\) is the eccentricity.
The distance between the two foci is \(2ae\).
3. Vertices: The points where the hyperbola intersects its transverse axis, located at \(V_1(a, 0)\) and \(V_2(-a, 0)\).
The distance between these two vertices is:
\[ \text{Distance} = a - (-a) = 2a \]
This distance, \(2a\), is defined as the length of the transverse axis.
Therefore, the length of the transverse axis is the distance between the two vertices.
This matches Option (A).

Step 4: Final Answer:
Therefore, the correct option is (A).