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All courses Math III · G-GPE.3.1 38 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 x²+y²=25?
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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

Inspect the squared terms

\[x^2+y^2=25\]

The equation has both variables squared and no xy term, so the signs and coefficients of x² and y² identify the conic type.

02

Compare the coefficients

\[1x^2+1y^2=25\]

The coefficient of x² is 1, and the coefficient of y² is also 1.

03

Check the signs

\[+x^2~\text{and}~+y^2\]

Both squared terms are positive because they are added on the left side.

04

Match the circle pattern

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

Equal positive coefficients on x² and y² match the pattern for a circle centered at the origin.

05

State the conic type

\[\text{conic}~\text{type}:~\text{circle}\]

Therefore, the conic represented by x²+y²=25 is a circle.

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Another way

  1. Check by rewriting the equation as x²+y²=5², so the radius is 5 and the center is (0,0). That is the standard circle pattern.

!

Common mistake

Calling x²+y²=25 an ellipse without noticing that the x² and y² coefficients are both 1. Equal positive coefficients make this a circle, not an ellipse.

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.