Problem preview

G-GPE.3.1 Warmup M3-038-A11-V04

Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.

Decide whether a conic equation is normal, degenerate, or has no real graph by inspecting squared terms and signs

Problem

Does \((x+3)^2/16+(y-1)^2/25=-2\) describe a normal conic, a degenerate conic, or no real graph?

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It has no real graph because both squared terms are nonnegative, so their sum cannot equal -2.

It is a degenerate hyperbola made of two intersecting lines because conics with unusual right sides break into lines.

It is a normal ellipse centered at (-3, 1) because both x and y are squared, added, and divided by positive numbers.

It is a degenerate conic that is just the point (-3, 1), because the ellipse collapses to its center.

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Curriculum context

Course: Math III
Standard: G-GPE.3.1
Category: Geometry
Domain: Expressing Geometric Properties with Equations
Objective: Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Problem type: Decide whether a conic equation is normal, degenerate, or has no real graph by inspecting squared terms and signs