SAT equation of a circle: the number on the right is r², not r
The equation ends in = 25. The radius is 5, the diameter is 10, and every one of those numbers is sitting in the choices waiting for you.
Why it happens
The standard form stores the radius squared, so every circle question has a hidden square root in it. Doubling a radius multiplies the right-hand side by four, not two — which is why "radius twice as large" questions have such convincing wrong answers.
See the trap
Circle P is (x + 4)² + (y − 1)² = 25. Circle Q has the same center and three times the radius. Write Q.
= 75, tripling the number that is written down.
r = 5, so Q has r = 15 and r² = 225. Tripling the radius multiplies the right-hand side by nine.
How to actually fix it
- 1Take the square root of the right-hand side immediately and write "r = …" in the margin.
- 2Do all the reasoning in terms of r, not r².
- 3Square once at the end, when you put the answer back into standard form.
Practice this trap · 4 real questions
Each of these is a question from the free bank that turns on the mistake above — the trap is the reason its wrong answer looks right. Easiest first.
FAQ
What if the equation is not in standard form?
If it arrives as x² + 8x + y² − 2y − 8 = 0 you have to complete the square first, and only then is the right-hand side r². Reading r² off an unfinished equation is the same trap one step earlier.
How do I stop repeating this mistake?
Reading the explanation isn't enough — your brain needs to beat the trap on a fresh question. Practice the skill in the kaoshen.co question bank: miss one and it is logged with the exact error, then brought back to you three days later and again after fourteen.
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