Circles are fundamental shapes in geometry, defined by points equidistant from a center. Their equations, derived from the Pythagorean theorem, allow us to precisely describe and analyze circular shapes on coordinate planes.

Understanding circle equations enables us to determine key features like center coordinates and radius length. This knowledge is crucial for graphing circles, finding their domains and ranges, and solving various geometric problems involving circular shapes.

Equations of Circles

Derivation of circle equation

• Derives the standard form of the equation of a circle using the Pythagorean theorem
  • Considers a circle with center coordinates $(h, k)$ and radius length $r$
  • Lets $(x, y)$ represent any arbitrary point on the circumference of the circle
  • Recognizes that the distance between the center $(h, k)$ and the point $(x, y)$ is always equal to the radius $r$
  • Applies the Pythagorean theorem to set up the equation $(x - h)^2 + (y - k)^2 = r^2$
    • $(x - h)$ represents the horizontal distance between $x$ and $h$
    • $(y - k)$ represents the vertical distance between $y$ and $k$
• Establishes the standard form of the equation of a circle as $(x - h)^2 + (y - k)^2 = r^2$
  • $h$ and $k$ represent the $x$ and $y$ coordinates of the center of the circle respectively
  • $r$ represents the length of the radius of the circle
  • Works for any circle on the coordinate plane

Writing circle equations

• Writes the equation of a circle using the standard form $(x - h)^2 + (y - k)^2 = r^2$
  • Substitutes the $x$ and $y$ coordinates of the center for $h$ and $k$ respectively
  • Substitutes the length of the radius for $r$
  • Expands and simplifies the equation if necessary
• Example: Given a circle with center $(-1, 3)$ and radius $2$, the equation would be $(x - (-1))^2 + (y - 3)^2 = 2^2$ which simplifies to $(x + 1)^2 + (y - 3)^2 = 4$
• Example: Given a circle with center $(0, 0)$ and radius $5$, the equation would be $(x - 0)^2 + (y - 0)^2 = 5^2$ which simplifies to $x^2 + y^2 = 25$

Center and radius from equations

• Determines the center coordinates and radius length of a circle from its equation in standard form
  • Compares the given equation to the standard form $(x - h)^2 + (y - k)^2 = r^2$
  • The $x$ and $y$ coordinates of the center $(h, k)$ are the values being subtracted from $x$ and $y$ in the equation
  • The radius $r$ is the square root of the constant term on the right side of the equation
• Example: Given the equation $(x - 4)^2 + (y + 2)^2 = 16$, the center is $(4, -2)$ and the radius is $\sqrt{16} = 4$
• Example: Given the equation $x^2 + y^2 - 6x + 4y - 12 = 0$, first rewrite in standard form:
  • $(x^2 - 6x) + (y^2 + 4y) = 12$
  • $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$
  • $(x - 3)^2 + (y + 2)^2 = 25$
  • The center is $(3, -2)$ and the radius is $\sqrt{25} = 5$

Graphing circles on coordinate plane

• Graphs a circle on the coordinate plane using its equation in standard form
  1. Identifies the center coordinates $(h, k)$ and radius length $r$ from the equation
  2. Plots the center point $(h, k)$ on the coordinate plane
  3. Plots four points $r$ units up, down, left, and right from the center
     • $(h, k + r)$, $(h, k - r)$, $(h - r, k)$, $(h + r, k)$
  4. Draws a smooth curve connecting these four points to form the circle
• The circle is symmetric about the vertical line $x = h$ and the horizontal line $y = k$ passing through the center
• The domain and range of a circle can be determined from its center and radius
  • Domain: All $x$ values from $h - r$ to $h + r$, or $[h - r, h + r]$
  • Range: All $y$ values from $k - r$ to $k + r$, or $[k - r, k + r]$
• Example: Given the equation $(x - 1)^2 + (y + 3)^2 = 4$
  • The center is $(1, -3)$ and the radius is $2$
  • The four points to plot are $(1, -1)$, $(1, -5)$, $(-1, -3)$, and $(3, -3)$
  • The domain is $[-1, 3]$ and the range is $[-5, -1]$