Problem 73 - College Algebra

Question

How can you distinguish parabolas from other conic sections by looking at their equations?

Step-by-Step Solution

Answer

A parabola can be distinguished from other conic sections by looking at their equations. A parabola's equation only includes one square term ('x^2' or 'y^2'), but never both. It also never contains an 'xy' term, unlike other conic sections. It's a second degree equation in one variable only, unlike ellipses and hyperbolas that are second degree equations in both variables.

Step 1: Understanding the Properties of a Parabola

A parabola has an equation of the form \(y=ax^2+bx+c\) or \(x=ay^2+by+c\) where \(a\), \(b\), and \(c\) are constants. The graph of a parabola always has a vertex and an axis of symmetry.

Step 2: Distinguishing Parabolas from Other Conic Sections

a) An ellipse (including circles) has an equation of the form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\). \ b) A hyperbola has an equation of the form \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\). If an equation matches to either of these forms, it's not a parabola. A crucial thing to note is that a parabolic equation has only one square power term, either \(x^2\) or \(y^2\), but never both as opposed to other conic sections.

Step 3: Practice on Examples

Applying the rules to identify conic sections to various equations and determining their type would help cement the understanding. We should remember that parabolas do not have an 'xy' term while hyperbolas and ellipses do.