ACT-Style: Circle in the Coordinate Plane

ACT Math · Axiom Academy

EXAMPLE ACT-Style: Circle in the Coordinate Plane Finding a circle's radius from its equation in general form, the way it actually shows up on the test A circle in the standard (x,y) coordinate plane has the equation x^2 + y^2 + 8x - 6y + 9 = 0 . What is the radius of the circle? A circle in general form is a completing-the-square problem wearing a disguise — the ACT's wrong answers are built from the specific slips that happen along the way. Complete the square on each variable separately: for x^2+bx , add ; the same move for y^2+cy . Here that's +16 for the x -group and +9 for the y -group. Standard form gives r^2 , not r : (x+4)^2+(y-3)^2=16 means r^2=16 — the radius is , not 16 itself. Radius diameter: 8 is the diameter ( 2r ) of this circle, a classic ACT trap answer for a radius question. Result: x^2+y^2+8x-6y+9=0 has radius 4 , answer choice C . On the ACT, circle-in-general-form questions are rarely wrong because of algebra mistakes elsewhere — they're wrong because the LAST step (square root vs. square, radius vs. diameter) gets skipped or flipped. Finish the derivation all the way to r , not just to r^2 .

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