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MathChapter 19: Circles

Notes

Equations of Circles Notes

Read and build circle equations, complete the square, and use coordinate geometry for diameters and axis tangency.

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About 48 minutes
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A circle equation stores two geometric facts: its center and radius. Standard form makes both visible; general form hides them until you complete the square.

Standard form

Equation of a circle
\[(x-h)^2+(y-k)^2=r^2\]

The center is \((h,k)\), and the radius is the positive square root of the right side.

Circle in standard formThe circle has equation (x-2)^2+(y+1)^2=16, center (2,-1), and radius 4. The graph window is x from -4 to 8 and y from -7 to 5, leaving a two-unit margin around the full circle. Purpose: read h, k, and r from standard form-4-22468-6-4-224xy(2, -1)center (2, -1), radius 4
Circle in standard form
Reading standard form
FeatureWhere it appearsCommon trap
Center x-coordinate hInside \((x-h)^2\)The visible sign is opposite h
Center y-coordinate kInside \((y-k)^2\)\((y+4)^2\) means \(k=-4\)
Radius r\(r=\sqrt{r^2}\)The right side is radius squared
Worked example

Write standard form

Write the equation of the circle with center \((-3,5)\) and radius 6.

  1. Substitute

    \((x-(-3))^2+(y-5)^2=6^2\).

  2. Simplify

    \((x+3)^2+(y-5)^2=36\).

\((x+3)^2+(y-5)^2=36\).

Complete the square

General-to-standard workflow

  1. Group

    Place x terms together, y terms together, and move the constant.

  2. Add twice

    For \(x^2+bx\), add \((b/2)^2\) to both sides; repeat for y.

  3. Factor

    Rewrite each trinomial as a binomial square, then read center and radius.

Worked example

Complete two squares

Find the center and radius of \(x^2+y^2-8x+6y-11=0\).

  1. Move constant

    \(x^2-8x+y^2+6y=11\).

  2. Add square terms

    Add 16 and 9 to both sides: \((x-4)^2+(y+3)^2=36\).

  3. Read

    Center \((4,-3)\); radius 6.

Center \((4,-3)\), radius 6.

Tangency, diameter endpoints, and area

Circle tangent to an axisThe circle has center (3,2), radius 3, and touches the y-axis at exactly one point. The graph window is x from -2 to 8 and y from -3 to 7, leaving a two-unit margin around the full circle. Purpose: use center-to-axis distance as radius-22468-2246xy(3, 2)center (3, 2), radius 3
Circle tangent to an axis
Circle from diameter endpointsThe diameter endpoints are (-4,1) and (2,1), so the midpoint is (-1,1) and the radius is 3. The graph window is x from -6 to 4 and y from -4 to 6, leaving a two-unit margin around the full circle. Purpose: use midpoint and half the endpoint distance-6-4-224-4-2246xy(-1, 1)center (-1, 1), radius 3
Circle from diameter endpoints
Worked example

Diameter endpoints

A diameter has endpoints \((-5,4)\) and \((3,-2)\). Find the center and \(r^2\).

  1. Midpoint

    The center is \((-1,1)\).

  2. Radius squared

    From \((-1,1)\) to \((3,-2)\), \(r^2=4^2+(-3)^2=25\).

Center \((-1,1)\), \(r^2=25\).
Mini check

Equation check

For \((x-2)^2+(y+7)^2=49\), state the center, radius, and area.

Show answer and explanation

Answer: Center \((2,-7)\), radius 7, area \(49\pi\).

Reverse the signs inside the binomials and use the positive square root of 49.

Key takeaways

Key takeaways

  • Standard form is \((x-h)^2+(y-k)^2=r^2\).
  • Complete a square by adding the square of half the linear coefficient to both sides.
  • Diameter endpoints give the center by midpoint and radius by half-distance.
  • Axis tangency turns a center coordinate’s absolute value into the radius.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.