Classify the Surface $$25 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$$

question 26

Multiple Choice

Classify the surface.
$$25 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 y + 2 z = 0$$
Select the correct answer.

• A) A hyperboloid of one sheet with center \(( - 3,0,1 )\) and axis parallel to the \(z\)-axis.
• B) A circular paraboloid with vertex \(( 0,3,1 )\) and axis the \(z\)-axis.
• C) A cone with axis parallel to the \(z\)-axis and vertex \(( 0,1,1 )\).

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Definitions:

Alternative Hypothesis

The hypothesis in a statistical test that proposes a specific difference or effect, in contrast to the null hypothesis which suggests no effect.

Rejection Region

The range of values in hypothesis testing that if the test statistic falls within, leads to the rejection of the null hypothesis.

Alternative Hypothesis

Represents a statement that contradicts the null hypothesis and is considered true if the null hypothesis is proven to be false, often denoting the presence of an effect or difference.

Test Statistic

A value calculated from sample data to evaluate a hypothesis in statistical hypothesis testing.