Problem 51
Question
Identify the quadric surface. $$ z^{2}=9 x^{2}+y^{2} $$
Step-by-Step Solution
Verified Answer
The given equation represents an elliptic cone.
1Step 1: Recognize the General Form of the Equation
Begin by identifying the general form of the given equation, which is of the form \(z^{2}=ax^{2}+by^{2}\), where a and b are any real numbers. This form matches the one for hyperboloids of one and two sheets and elliptic cones.
2Step 2: Identify the Constants
Next, identify the values of a and b in the given equation. Here, a is 9 and b is 1.
3Step 3: Identify the Quadric Surface
Notice that a is larger than b. This situation corresponds to a hyperboloid of one sheet when a, b > 0; a hyperboloid of two sheets when a, b < 0; or an elliptic cone when either a or b is zero. In this case, both a and b are greater than zero, and a is larger than b. Hence, the given equation represents an elliptic cone.
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