How to handle circle equations on the SAT
Read centre and radius straight from standard form, convert from expanded form, and avoid the two sign traps the SAT uses.
What the examiner is testing
Whether you can move between the two forms of a circle equation. The SAT rarely asks you to draw one — it asks for the centre, the radius, or the diameter, and hides the answer one algebraic step away.
$$ (x - h)^2 + (y - k)^2 = r^2 $$
Centre \( (h, k) \), radius \( r \). Note that the equation stores \( r^2 \), not \( r \) — the single most common slip on this topic.
Reading it straight off
For \( (x - 3)^2 + (y + 5)^2 = 49 \):
- \( h = 3 \), because the bracket says \( x - 3 \).
- \( k = -5 \), because \( y + 5 \) is \( y - (-5) \).
- \( r = \sqrt{49} = 7 \), not 49.
Centre \( (3, -5) \), radius 7. The signs flip. A bracket reading \( +5 \) means a coordinate of \( -5 \).
Worked example: converting from expanded form
A circle is given by \( x^2 + y^2 - 6x + 4y - 12 = 0 \). Find its centre and radius.
Step 1 — group x terms and y terms, move the constant right.
$$ (x^2 - 6x) + (y^2 + 4y) = 12 $$
Step 2 — complete the square on the x group. Half of \( -6 \) is \( -3 \), and \( (x-3)^2 \) introduces \( +9 \):
$$ x^2 - 6x = (x - 3)^2 - 9 $$
Step 3 — complete the square on the y group. Half of \( 4 \) is \( 2 \), and \( (y+2)^2 \) introduces \( +4 \):
$$ y^2 + 4y = (y + 2)^2 - 4 $$
Step 4 — substitute both back.
$$ (x - 3)^2 - 9 + (y + 2)^2 - 4 = 12 $$
Step 5 — move the constants to the right.
$$ (x - 3)^2 + (y + 2)^2 = 25 $$
Answer: centre \( (3, -2) \), radius \( \sqrt{25} = 5 \). If the question asks for the diameter, it is 10 — read the question twice, because both appear as answer choices.
Check: expand back mentally. \( (x-3)^2 + (y+2)^2 = x^2 - 6x + 9 + y^2 + 4y + 4 \), which equals 25, so \( x^2 + y^2 - 6x + 4y = 12 \). That matches the original.
The three mistakes that lose marks
1. Giving \( r^2 \) as the radius. The right-hand side is the radius squared. An answer of 25 when the radius is 5 is a distractor the SAT deliberately offers.
2. Copying the sign from the bracket. \( (y + 2)^2 \) means \( k = -2 \). Both signs flip, every time.
3. Forgetting to move the introduced constants. Completing the square adds a number you must subtract again. Track them on the left before shifting them right.
30-second recap
Standard form gives centre and radius directly, with the signs reversed and the radius square-rooted. To get there from expanded form: group, complete the square on each variable, move every constant to the right. Then check whether the question wanted radius or diameter.