Problem 81 - Algebra and Trigonometry

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 its equation. If the equation has only one squared term, either \(x^2\) or \(y^2\), and no xy term, it is a parabola. All other conic sections' equations will include both squared terms.

Step 1: Identifying Conic Sections

Firstly, we need to understand that a conic section's equation is typically represented in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, A, B, C, D, E, and F are constants. The type of conic section the equation represents (ellipse, parabola, hyperbola, or circle) is determined by the values and interactions of A, B and C.

Step 2: Characteristics of Parabolas

In the case of parabolas, one of the coefficients A or C is zero and B equals zero. Therefore the standard form of a parabola's equation can either be \(Ax^2 + Dx + Ey + F = 0\) or \(Cy^2 + Dx + Ey + F = 0\). This means that a parabola is the only conic section whose equation is not represented by a combination of both squared terms \(x^2\) and \(y^2\). Hence, if you see an equation with only one squared term, it is a parabola.

Step 3: Characteristics of Other Conic Sections

For other conic sections, both the coefficients A and C exist, and the value of B determines the type of the conic section. If \(B^2 - 4AC = 0\), the conic is a circle. If \(B^2 - 4AC < 0\), the conic is an ellipse. And if \(B^2 - 4AC > 0\), the conic is a hyperbola.