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All courses Algebra II · G-GPE.3.1 44 of 55
Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.

Identify the type of conic by inspecting squared terms and signs in the equation

Problem
What type of conic is represented by \(x^2+y^2=25\)?
Your answer
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Hint

A circle has x squared and y squared with the same coefficient and the same sign.

Here both squared terms have coefficient 1 and are added, so the conic is a circle.

Solution walkthrough

01

Compare the squared-variable terms

\[x^2~\text{and}~y^2~\text{have}~\text{equal}~\text{positive}~\text{coefficients}~1\]

A circle has both squared variables with the same positive coefficient and no xy term.

02

Match standard form

\[x^2+y^2=r^2\]

The equation already matches a circle centered at the origin.

03

Read the radius as a check

\[r^2=25~->~r=5\]

The positive radius is 5, confirming that the locus is a nondegenerate circle.

04

Classify the conic

\[\text{It}~\text{is}~a~\text{circle}~\text{because}~x^2~\text{and}~y^2~\text{have}~\text{the}~\text{same}~\text{positive}~\text{coefficient},~\text{matching}~\text{the}~\text{form}~x^2+y^2=r^2.\]

The coefficient pattern and standard-form match provide the requested justification.

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Another way

  1. Solve for points at the axes: (plus or minus 5,0) and (0,plus or minus 5) are all equally distant from the origin, revealing a circle.

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Common mistake

Do not call the conic an ellipse merely because a circle belongs to the ellipse family. Equal squared-term coefficients identify the more specific classification circle.

Equal-scale circle x squared plus y squared equals 25 with center, radius, and coefficient-based classification.
Equal positive squared coefficients produce a circle centered at the origin with radius 5.

Solution walkthrough video