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Two branches of a hyperbola...

Two branches of a hyperbola

A

a) have a common tangent

B

b) have a common normal

C

c) do not have a common tangent

D

d) do not have a common normal

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The correct Answer is:
To solve the question regarding the properties of the branches of a hyperbola, we will analyze the options provided and determine which statements are true based on the characteristics of hyperbolas. ### Step-by-Step Solution: 1. **Understanding Hyperbola Properties**: - A hyperbola consists of two separate branches that open in opposite directions. - The branches of a hyperbola have specific geometric properties regarding tangents and normals. 2. **Analyzing Common Tangents**: - For two branches of a hyperbola, it is known that they do not share a common tangent. This is due to the fact that tangents to each branch will not intersect at a point that lies on both branches simultaneously. 3. **Analyzing Common Normals**: - In contrast, the branches of a hyperbola can have a common normal. A normal is a line perpendicular to the tangent at a given point on the curve. Since the branches are separate, it is possible for them to have a normal line that is common to both branches at different points. 4. **Evaluating the Options**: - **Option A**: Have a common tangent - **False** (as established, hyperbola branches do not share a common tangent). - **Option B**: Have a common normal - **True** (the branches can have a common normal). - **Option C**: Do not have a common tangent - **True** (as established). - **Option D**: Do not have a common normal - **False** (as established, they can have a common normal). 5. **Final Conclusion**: - Therefore, the correct options are: - Option B: Have a common normal. - Option C: Do not have a common tangent. ### Final Answer: Thus, the correct options are: - Option B: Have a common normal. - Option C: Do not have a common tangent. ---
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