Circles in the Coordinate Plane

TipLearning Objectives
  • Identify center and radius from circle equations.
  • Interpret geometric features of circles on the coordinate plane.
  • Convert between standard and expanded forms.

Key Ideas

Standard Form

\[ (x-h)^2 + (y-k)^2 = r^2 \]

  • Center \((h,k)\)
  • Radius \(r\)

Expanded Form

Complete the square to convert to standard form.

Circle on the coordinate plane with center labeled \((h, k)\).

Common Problem Types

Identifying Center and Radius

From \((x-h)^2+(y-k)^2=r^2\).

Converting Expanded → Standard

Complete the square for \(x\)-terms and \(y\)-terms.

Check Whether a Point Lies on Circle

Plug into equation.

Circle–Line Intersection (Basic SAT)

Substitution to check if point is inside, on, or outside.

Strategies

  • Move constants to one side before completing square.
  • Group \(x\)-terms and \(y\)-terms together.
  • Square half the coefficient.

Worked Examples

Example 1

Find center and radius:
\((x-3)^2 + (y+2)^2 = 49\)

Center = \((3,-2)\)
Radius = \(7\)

Example 2

Convert to standard:
\(x^2 + y^2 - 4x + 6y + 9 = 0\)

Group terms:
\((x^2 - 4x) + (y^2 + 6y) = -9\)

Complete squares:
\((x-2)^2 - 4 + (y+3)^2 - 9 = -9\)

Move constants:
\((x-2)^2 + (y+3)^2 = 4\)

Radius \(=2\).

WarningCommon Mistakes
  • Forgetting to add square-completion constants to both sides.
  • Incorrectly identifying signs of center.
  • Using radius instead of radius-squared.

Practice Problems

  1. Find center & radius: \((x+1)^2+(y-5)^2=64\).
  2. Convert: \(x^2 + y^2 + 8x - 4y - 5 = 0.\)
  3. Does (3,2) lie on \((x-3)^2+(y-2)^2=0\)?
  1. Center = \((-1,5)\), radius = 8.
  2. Standard: \((x+4)^2 + (y-2)^2 = 25.\)
  3. Substitution gives 0 → yes, it is the center (degenerate circle).

Summary

  • Standard form reveals center and radius.
  • Complete square for conversions.
  • Use substitution to test points.
  • Flip the sign inside parentheses to get the center.
  • Radius is \(\sqrt{r^2}\), not \(r^2\).