Recognizing Circle Equations

SAT Glossary

An equation with x^2 and y^2 at equal coefficients and no xy term is always a circle; complete the square in both variables to recover centre and radius. The named distractor takes √() of the loose constant — reporting 3 from x^2+y^2-6x+8y+9=0 — the reward for never completing the square.

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term mixing them together, that equation is secretly describing a CIRCLE — even if it doesn't look like one yet. To reveal the circle's actual center and radius, you complete the square separately for the x-terms and the y-terms. The classic trap: some students skip completing the square entirely and just take the square root of whatever loose constant is sitting in the equation — for example, wrongly reporting a radius of 3 straight from the number in $x^2+y^2-6x+8y+9=0$, when the real radius only reveals itself correctly after actually completing the square on both variables.","aliases":[],"topic":"Course Foundations","priority":"specialized","learningStage":"intermediate","prerequisiteIds":[],"questionCount":0,"curriculumBasis":{"revision":"current-sat-2026","objective":"Course Foundations: The circle in disguise","source":"https://satsuite.collegeboard.org/practice/content-domains"},"id":"Recognizing Circle Equations","courseId":"sat","publicSeo":true,"seoSlug":"recognizing-circle-equations-x16a47"}

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Recognizing Circle Equations

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