Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ x^{2}+\frac{y^{2}}{4}+z^{2}=1 $$

Step-by-Step Solution

Answer

The quadric surface represented by the given equation \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\) is an ellipsoid.

Step 1: Analyze the given equation

The provided equation is \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\). As a quadric surface, we know this value must represent a three-dimensional figure. All terms involving variables (x, y and z) are squared in this equation, thus we should look at those types of quadric surfaces.

Step 2: Compare the equation to the standard forms of quadric surfaces

Considering different types of quadric surfaces and their standard forms: \n 1. Ellipsoid: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) \n Here, a, b, and c are all positive constants, and a, b, and c are all non-zero. \n 2. Hyperboloid: \(x^{2}/a^{2}±y^{2}/b^{2}-z^{2}/c^{2}=1\) \n 3. Paraboloid: \(x^{2}/a^{2}±y^{2}/b^{2}=z/c\) \n \n Our given equation \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\) matches the form of the ellipsoid, with \(a^{2}=1\), \(b^{2}=4\), and \(c^{2}=1\). Thus, this equation describes an ellipsoid.

Step 3: Confirm the description of the surface

The final step in the solution involves confirming that the given equation indeed describes an ellipsoid. For this equation, a, b, and c (whenever exist) satisfy a > 0, b > 0, and c > 0, and thus describe an ellipsoid. The values for a, b, and c impact the 'stretch' of the ellipsoid along the x, y, and z axes respectively. As the equation \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\) satisfies this condition, this confirms that the given equation describes an ellipsoid.