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Circles: standard form and general form

Level: 101 · for anyone starting precalculus

One line: A circle is every point at one distance from one point, so its equation is the distance formula held fixed, (x − h)² + (y − k)² = r²; the general form hides the center and radius, completing the square brings them back, and when the number that should be r² comes out zero or negative there is no circle at all.

The idea

A circle is the set of all points in the plane at a fixed distance r, the radius, from a fixed point (h, k), the center. Read with the distance formula, a point (x, y) is on the circle exactly when √((x − h)² + (y − k)²) = r. Square both sides, which is safe because both are non-negative, and that is the equation:

(x − h)² + (y − k)² = r², the standard form.

The center and radius are right there to read off, with one trap: the signs. (x + 2)² is (x − (−2))², so its center has x = −2. The unit circle, center (0, 0) and radius 1, is x² + y² = 1, and it is where trigonometry will live.

Multiply out the squares and gather everything on one side, and the equation becomes x² + y² + ax + by + c = 0, the general form. It is what a circle looks like when it turns up in a problem, and the center and radius are no longer visible. To get them back, complete the square in x and in y. And if what comes out on the right is 0, the "circle" is one point; if it is negative, there are no points at all, since two squares cannot add up to less than 0.

This page follows section 1.4, Circles, in Michael Sullivan's Precalculus, the last section of the book's first chapter.

Symbol by symbol

Piece Say it What it means
(h, k) "h, k" the center. The letters are a convention; they are just the center's x and y.
r "r" the radius, a length, so r > 0. The equation has r², so read the radius as the square root of the right side.
(x − h)² the squared horizontal distance from the center.
standard form (x − h)² + (y − k)² = r², center and radius visible.
general form x² + y² + ax + by + c = 0, multiplied out. x² and y² always have the same coefficient in a circle.
complete the square x² + ax = (x + a/2)² − (a/2)²: add (a/2)², half the coefficient squared, to make a perfect square, and add the same to the other side.

Completing the square, step by step

Take x² + y² + 4x − 6y + 12 = 0.

  1. Group the x's and the y's, move the number across: (x² + 4x) + (y² − 6y) = −12.
  2. Complete each square: half of 4 is 2, squared 4; half of −6 is −3, squared 9. Add both to both sides: (x² + 4x + 4) + (y² − 6y + 9) = −12 + 4 + 9.
  3. Factor: (x + 2)² + (y − 3)² = 1.

Center (−2, 3), radius 1. Change the 12 to 13 and the right side becomes 0: only the point (−2, 3). Change it to 14 and the right side is −1: no points. The program runs all three.

How can a continuous thing be a set?

The definition says a circle is a set of points, and the owner's objection is fair: a circle looks like one continuous thing, and a set sounds like a bag of separate things. The answer is in two parts, and the program's section 8 runs both.

First, "set of points" means membership by test, exactly as in what is a set?. The unit circle is {(x, y) : x² + y² = 1}, a set-builder expression, and a point is in it or not according to whether its coordinates pass the test: (3/5, 4/5) passes, (1/2, 1/2) fails. Nothing is drawn; the circle is the collection of every pair that passes. That is the whole content of Sullivan's sentence, and it is why one coordinate system turns geometric statements into algebraic ones: a figure is the solution set of its equation.

Second, "continuous" is a property of that set, not something a set lacks. Between any two points of the circle there is a third (the program bisects the parameter five times and finds a new point each time, with exact fractions), so no point has a next point and there are no gaps; every limit of points of the circle is still on it; and the set is one piece, in that it cannot be split into two nonempty parts neither of which comes arbitrarily close to the other. Each of these is a sentence about which points are members, which is all a set is. The dense rational points, one for every fraction t, show the set is not a finite list of dots, and the real parameter t shows it is uncountable, with as many points as an interval. The idea that a line is a set of points, with its continuity captured by a property of the set, was Dedekind's in 1872, and it is the moment geometry became part of set theory; the reading guide places it among the branches. What a set of points can do once it is only a set, and no longer a shape, is the subject of two balls from one.

What the program prints

The picture marks every whole-number point exactly 5 from the origin. There are twelve, and each is the corner of a 3-4-5 (or 0-5-5) right triangle, which is the Pythagorean theorem again. Inside, on, or outside is decided by comparing the squared distance with r², with no square roots.

Verified output of circles.py — regenerated by tools/run_examples.py, never hand-typed.

1. A CIRCLE IS ONE DISTANCE FROM ONE POINT: x^2 + y^2 = 25
     6   .  .  .  .  .  .  |  .  .  .  .  .  .
     5   .  .  .  .  .  .  *  .  .  .  .  .  .
     4   .  .  .  *  .  .  |  .  .  *  .  .  .
     3   .  .  *  .  .  .  |  .  .  .  *  .  .
     2   .  .  .  .  .  .  |  .  .  .  .  .  .
     1   .  .  .  .  .  .  |  .  .  .  .  .  .
     0   -  *  -  -  -  -  C  -  -  -  -  *  -
    -1   .  .  .  .  .  .  |  .  .  .  .  .  .
    -2   .  .  .  .  .  .  |  .  .  .  .  .  .
    -3   .  .  *  .  .  .  |  .  .  .  *  .  .
    -4   .  .  .  *  .  .  |  .  .  *  .  .  .
    -5   .  .  .  .  .  .  *  .  .  .  .  .  .
    -6   .  .  .  .  .  .  |  .  .  .  .  .  .
        -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6   x
   12 whole-number points are exactly 5 from the origin:
   (-5,0) (-4,-3) (-4,3) (-3,-4) (-3,4) (0,-5) (0,5) (3,-4) (3,4) (4,-3) (4,3) (5,0)
   They come from 3-4-5 and 0-5-5: each is a right triangle's hypotenuse of 5.

2. STANDARD FORM FROM CENTER AND RADIUS, THEN THE GENERAL FORM
   center (0, 0), r = 1:
      x^2 + y^2 = 1                   ->  x^2 + y^2 - 1 = 0
   center (-2, 3), r = 4:
      (x + 2)^2 + (y - 3)^2 = 16      ->  x^2 + y^2 + 4x - 6y - 3 = 0
   center (3, -1), r = 1/2:
      (x - 3)^2 + (y + 1)^2 = 1/4     ->  x^2 + y^2 - 6x + 2y + 39/4 = 0
   The sign flips: center (-2, 3) is written (x + 2) and (y - 3).

3. BACK AGAIN: COMPLETE THE SQUARE
   x^2 + y^2 + 4x - 6y + 12 = 0
   group:     (x^2 + 4x) + (y^2 - 6y) = -12
   complete:  (x^2 + 4x + 4) + (y^2 - 6y + 9) = -12 + 4 + 9    add (half the coefficient)^2
   so:        (x + 2)^2 + (y - 3)^2 = 1
   center (-2, 3), radius 1. Whatever is added on the left is added on the right.

4. NOT EVERY x^2 + y^2 + ax + by + c = 0 IS A CIRCLE
   x^2 + y^2 + 4x - 6y + 12 = 0  a circle: (x + 2)^2 + (y - 3)^2 = 1, center (-2, 3), radius 1
   x^2 + y^2 + 4x - 6y + 13 = 0  only the point (-2, 3): (x + 2)^2 + (y - 3)^2 = 0, radius 0
   x^2 + y^2 + 4x - 6y + 14 = 0  no points at all: (x + 2)^2 + (y - 3)^2 = -1, and a sum of squares is never -1
   x^2 + y^2 - x = 0             a circle: (x - 1/2)^2 + y^2 = 1/4, center (1/2, 0), radius 1/2
   The right side after completing the square is r^2. Positive: a circle.
   Zero: a single point. Negative: nothing, since squares cannot add to less than 0.

5. INSIDE, ON OR OUTSIDE? COMPARE THE SQUARED DISTANCE WITH r^2
   circle (x + 2)^2 + (y - 3)^2 = 16
   ( 2, 3):  (x + 2)^2 + (y - 3)^2 = 16   on
   ( 0, 0):  (x + 2)^2 + (y - 3)^2 = 13   inside
   ( 1, 6):  (x + 2)^2 + (y - 3)^2 = 18   outside
   (-2, 3):  (x + 2)^2 + (y - 3)^2 =  0   inside
   ( 3, 5):  (x + 2)^2 + (y - 3)^2 = 29   outside

6. INTERCEPTS OF A CIRCLE: (x - 1)^2 + (y + 2)^2 = 5
   y = 0: (x - 1)^2 + 4 = 5, (x - 1)^2 = 1, x = 0, 2
   x = 0: 1 + (y + 2)^2 = 5, (y + 2)^2 = 4, y = -4, 0
   0 is in both lists, so the circle passes through the origin, since 1 + 4 = 5.
   A circle can meet an axis twice, once (touching it), or never.

7. THE UNIT CIRCLE: CENTER (0, 0), RADIUS 1, x^2 + y^2 = 1
   (    1,     0):  x^2 + y^2 = 1
   (  3/5,   4/5):  x^2 + y^2 = 1
   (-5/13, 12/13):  x^2 + y^2 = 1
   (    0,    -1):  x^2 + y^2 = 1
   Every Pythagorean triple a, b, c gives the point (a/c, b/c) on it.
   Trigonometry will name its points (cos t, sin t).

8. A CIRCLE IS A SET OF POINTS, AND A CONTINUOUS ONE
   membership is a test, like any set builder: (x, y) ∈ circle iff x^2 + y^2 = 1
      (3/5, 4/5) ∈ unit circle: True
      (1/2, 1/2) ∈ unit circle: False
      (0, 1) ∈ unit circle: True
   a rational point for every fraction t: ((1 - t^2)/(1 + t^2), 2t/(1 + t^2))
      13 distinct points with t = m/n, n ≤ 6, all on the circle: True
   between any two points of the circle there is a third, so no point has a
   next point; the fractions t do that on their own:
      t = 1/2: x = 3/5 lies between the x's for t = 0 and t = 1
      t = 1/4: x = 15/17 lies between the x's for t = 0 and t = 1/2
      t = 1/8: x = 63/65 lies between the x's for t = 0 and t = 1/4
      t = 1/16: x = 255/257 lies between the x's for t = 0 and t = 1/8
      t = 1/32: x = 1023/1025 lies between the x's for t = 0 and t = 1/16
   'Continuous' is not a feeling about the drawing but two facts about the set:
   no gaps (between any two points a third, as above, and every limit of points
   of the circle is on the circle) and one piece (it cannot be split into two
   nonempty parts with no point of either touching the other). Both are
   statements about which points are members. The set is uncountable: t runs
   through all reals in [0, 1] and so cannot be listed (see cardinality).

To check whether a general-form equation is a circle, give its five coefficients, for A x² + B y² + a x + b y + c = 0:

python3 08_Analytic_Geometry/circles/examples/circles.py 1 1 -4 6 -3
python3 08_Analytic_Geometry/circles/examples/circles.py 2 2 -12 8 -24

Questions

1. Write the standard form of the circle with center (−2, 3) and radius 4. Then the general form.

Answer (x + 2)² + (y − 3)² = 16. Expanded: x² + 4x + 4 + y² − 6y + 9 = 16, so x² + y² + 4x − 6y − 3 = 0.

2. Find the center and radius of (x − 3)² + (y + 1)² = 7.

Answer Center (3, −1), radius √7. Not 7: the right side is r².

3. Find the center and radius of x² + y² − 4x + 6y − 3 = 0.

Answer (x² − 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9 = 16, so (x − 2)² + (y + 3)² = 16: center (2, −3), radius 4.

4. Find the center and radius of 2x² + 2y² − 12x + 8y − 24 = 0.

Answer Divide by 2 first: x² + y² − 6x + 4y − 12 = 0. Then (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25: center (3, −2), radius 5. Completing the square only works with x² and y² having coefficient 1.

5. Is x² + y² + 4x − 6y + 14 = 0 a circle?

Answer No. Completing the square gives (x + 2)² + (y − 3)² = −1, and a sum of squares is never negative. The graph is empty.

6. Is x² + 2y² = 4 a circle?

Answer No: x² and y² have different coefficients. It is an ellipse, stretched along the x-axis, with intercepts ±2 and ±√2.

7. Find the intercepts of (x − 1)² + (y + 2)² = 5.

Answer y = 0: (x − 1)² + 4 = 5, x − 1 = ±1, x = 0 or 2. x = 0: 1 + (y + 2)² = 5, y + 2 = ±2, y = 0 or −4. 0 is in both lists: the circle passes through the origin.

8. Find the equation of the circle with a diameter from (1, 4) to (−3, 2).

Answer The center is the [midpoint](../midpoint_formula/README.md), (−1, 3). The radius is the distance from the center to an end: √(4 + 1) = √5. So (x + 1)² + (y − 3)² = 5.

9. The circle has center (2, −3) and passes through (5, 1). Find its equation.

Answer r² = (5 − 2)² + (1 + 3)² = 9 + 16 = 25. (x − 2)² + (y + 3)² = 25. No square root needed, because the equation wants r².

10. Is (0, 0) inside, on or outside the circle (x + 2)² + (y − 3)² = 16?

Answer (0 + 2)² + (0 − 3)² = 13 < 16: inside.

11. Which symmetries does every circle centered at the origin have? Every circle centered at (2, −3)?

Answer x² + y² = r² passes all three tests of [graphs, intercepts and symmetry](../graphs_intercepts_symmetry/README.md): x-axis, y-axis and origin. A circle centered at (2, −3) is symmetric about the vertical line x = 2, the horizontal line y = −3, and its center, but in general about none of the axes.

12. Why can (3/5, 4/5) be written down as a point of the unit circle without a calculator?

Answer (3/5)² + (4/5)² = 9/25 + 16/25 = 1. It is the 3-4-5 triangle scaled by 1/5. Every Pythagorean triple a, b, c gives the point (a/c, b/c).

How to practise

  1. Read center and radius aloud with the sign flip. (x + 2)² + (y − 3)² = 7: "center minus two, three; radius root seven."
  2. Completing the square, all three steps written out. Group, add half-squared to both sides, factor. Check the result by expanding it again.
  3. Divide first when x² and y² carry a coefficient.
  4. Check with the program, and do the cards.

Flashcards

The page as a deck of Anki cards: circles.txt. Import with File → Import. Tags: definition, standard-form, general-form, problems, trap.

Where this goes next

This is the end of the first chapter of the book. Chapter 2 is functions, and the circle is the first example of a graph that is not the graph of a function, since a vertical line meets it twice. In trigonometry the unit circle becomes the definition of sine and cosine, (cos t, sin t), and in this library it is already where roots of unity sit and what radians measure.

Po polsku, w skrócie

Okrąg to zbiór punktów odległych o r od środka (h, k). Wzór na odległość, podniesiony do kwadratu, daje od razu równanie w postaci kanonicznej: (x − h)² + (y − k)² = r². Pułapka ze znakami: (x + 2)² znaczy, że środek ma x = −2; a po prawej stronie jest r², więc promień to pierwiastek z tej liczby.

Po wymnożeniu dostajemy postać ogólną x² + y² + ax + by + c = 0, w której środka i promienia nie widać. Odzyskuje się je, uzupełniając do kwadratu: do x² + ax dodajemy (a/2)², do obu stron równania. Jeśli po prawej wyjdzie liczba dodatnia, to okrąg; zero, to jeden punkt; ujemna, to nic, bo suma kwadratów nie jest ujemna. Gdy przy x² i y² stoi ta sama liczba różna od 1, najpierw przez nią dzielimy; gdy różne liczby, to nie okrąg, tylko elipsa. Program rozstrzyga to dokładnie dla dowolnych współczynników.

Jak coś ciągłego może być zbiorem? Okrąg to zbiór par (x, y), które przechodzą test x² + y² = 1; ciągłość nie jest cechą rysunku, lecz tego zbioru: między każdymi dwoma punktami leży trzeci, każda granica punktów okręgu leży na okręgu, i zbiór jest jednym kawałkiem. Program sprawdza to na dokładnych ułamkach: dla każdego t = m/n punkt ((1 − t²)/(1 + t²), 2t/(1 + t²)) leży na okręgu, a dzielenie t na pół daje wciąż nowe punkty. Punktów jest nieprzeliczalnie wiele, tyle co liczb rzeczywistych w przedziale.

Auf Deutsch: Stichwörter

Ein Kreis in Normalform (x − a)² + (y − b)² = r² und in allgemeiner Form; quadratische Ergänzung macht aus der einen die andere.

Stichwörter: Kreisgleichung, Mittelpunkt, Radius, Normalform, allgemeine Form, quadratische Ergänzung, Einheitskreis.

See also