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#deck:Math::Circles
#tags:math precalculus circles
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What is a circle?	The set of all points at a fixed distance r, the radius, from a fixed point, the center.	definition
Standard form of a circle with center (h, k) and radius r?	(x - h)^2 + (y - k)^2 = r^2.	standard-form
Where does the standard form come from?	The distance formula from (x, y) to (h, k), set equal to r and squared.	standard-form
The unit circle's equation?	x^2 + y^2 = 1: center (0, 0), radius 1.	standard-form
Center and radius of (x - 3)^2 + (y + 1)^2 = 7?	Center (3, -1), radius the square root of 7.	problems
Center and radius of x^2 + (y - 2)^2 = 9?	Center (0, 2), radius 3.	problems
Standard form, center (-2, 3), radius 4?	(x + 2)^2 + (y - 3)^2 = 16.	problems
General form of a circle?	x^2 + y^2 + ax + by + c = 0, with x^2 and y^2 having the same coefficient.	general-form
How do you get center and radius from the general form?	Complete the square in x and in y.	general-form
How do you complete the square on x^2 + ax?	Add (a/2)^2: x^2 + ax + (a/2)^2 = (x + a/2)^2. Add the same to the other side.	general-form
x^2 + y^2 + 4x - 6y + 12 = 0: center and radius?	(x + 2)^2 + (y - 3)^2 = 1: center (-2, 3), radius 1.	problems
x^2 + y^2 - 4x + 6y - 3 = 0: center and radius?	(x - 2)^2 + (y + 3)^2 = 16: center (2, -3), radius 4.	problems
2x^2 + 2y^2 - 12x + 8y - 24 = 0: first step?	Divide by 2. Then center (3, -2), radius 5.	problems
After completing the square the right side is 0. What is the graph?	A single point, the center.	general-form
After completing the square the right side is negative. What is the graph?	Nothing: a sum of squares is never negative.	general-form
Is x^2 + 2y^2 = 4 a circle?	No: different coefficients on x^2 and y^2. It is an ellipse.	general-form
How do you tell if a point is inside a circle?	Its squared distance from the center is less than r^2.	problems
Circle with a diameter from (1, 4) to (-3, 2)?	Center the midpoint (-1, 3), r^2 = 5: (x + 1)^2 + (y - 3)^2 = 5.	problems
Circle with center (2, -3) through (5, 1)?	r^2 = 9 + 16 = 25: (x - 2)^2 + (y + 3)^2 = 25.	problems
How many whole-number points lie on x^2 + y^2 = 25?	12: the four on the axes and eight from 3-4-5.	problems
Why is (3/5, 4/5) on the unit circle?	9/25 + 16/25 = 1: the 3-4-5 triangle scaled by 1/5.	problems
How many times can a circle meet the x-axis?	Twice, once (touching), or never.	problems
Is a circle the graph of a function y = f(x)?	No. A vertical line through it meets it twice.	definition
Trap: (x + 2)^2 + (y - 3)^2 = 16 has center (2, -3)?	No, (-2, 3). x + 2 = x - (-2).	trap
Trap: (x - 1)^2 + (y - 1)^2 = 9 has radius 9?	No, 3. The right side is r^2.	trap
Trap: completing x^2 + 6x, add 6^2 = 36?	No. Add half of 6, squared: 9.	trap
Trap: complete the square on the left, leave the right side alone?	No. Whatever is added on the left must be added on the right.	trap
Trap: x^2 + y^2 + 4x - 6y + 14 = 0 is a small circle?	No. It becomes (x + 2)^2 + (y - 3)^2 = -1: no points at all.	trap
Trap: 3x^2 + 3y^2 - 6x = 0: complete the square on x^2 - 6x?	Divide by 3 first: x^2 + y^2 - 2x = 0, then (x - 1)^2 + y^2 = 1.	trap
Trap: to find r from the center and a point, take the square root and then square it again?	Unneeded. The equation wants r^2, which is the squared distance itself.	trap