Question:
Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
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For quadratic forms in Euclidean spaces, if the equation represents a hyperboloid of one sheet, the set is connected. If it represents a hyperboloid of two sheets, the set is disconnected.
Updated On: Apr 9, 2025
- \( S \) is connected, but \( T \) and \( U \) are not connected.
- \( T \) and \( U \) are connected, but \( S \) is not connected.
- \( S \) and \( U \) are connected, but \( T \) is not connected.
- \( S \) and \( T \) are connected, but \( U \) is not connected.
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The Correct Option is D
Solution and Explanation
Set \( S \): The equation \( x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \) represents a cone in \( \mathbb{R}^4 \). This cone is connected because we can travel continuously along it.
Set \( T \): The equation \( x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \) represents a hyperboloid of one sheet in \( \mathbb{R}^4 \). This set is connected as it forms a continuous surface.
Set \( U \): The equation \( x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \) represents a hyperboloid of two sheets in \( \mathbb{R}^4 \), which is disconnected because it consists of two disjoint components.
Thus, \( S \) and \( T \) are connected, but \( U \) is not connected.
\[ \boxed{D} \quad S \text{ and } T \text{ are connected, but } U \text{ is not connected.} \]
Set \( T \): The equation \( x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \) represents a hyperboloid of one sheet in \( \mathbb{R}^4 \). This set is connected as it forms a continuous surface.
Set \( U \): The equation \( x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \) represents a hyperboloid of two sheets in \( \mathbb{R}^4 \), which is disconnected because it consists of two disjoint components.
Thus, \( S \) and \( T \) are connected, but \( U \) is not connected.
\[ \boxed{D} \quad S \text{ and } T \text{ are connected, but } U \text{ is not connected.} \]
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