Question

the conic section below is________. a. a circle b. an ellipse c. a parabola d. a hyperbola e. a degenerate

Explanation:

Step1: Recall the properties of conic sections

A circle has \(x^{2}+y^{2}=r^{2}\) (all points at equal distance from center). An ellipse has \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (\(a
eq b\)). A parabola has \(y=ax^{2}+bx + c\) (U - shaped or sideways U - shaped). A hyperbola has \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) (two - part curve). A degenerate conic is a special case like a point, line, or two intersecting lines.

Step2: Analyze the given graph

The given graph is a closed curve, symmetric about \(x\) - axis and \(y\) - axis. It is not a circle (since it is elongated along \(x\) - axis). It is not a parabola (not U - shaped). It is not a hyperbola (not two - part). It is not degenerate. It satisfies the general form of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a>b\)) as it is a closed curve symmetric about both axes with different semi - major and semi - minor axes.

Answer:

B. an ellipse