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Grade 11 · lesson 1 of 4

Read a Circle’s Center and Radius

The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Identify center and radius of (x−2)²+(y+3)²=4. Candidate answers: (A) Center (2,3); radius 2 (B) Center (2,−3); radius 2 (C) Center (2,−3); radius 4 (D) Center (−2,3); radius 2.
  • Independent check inputs: Identify center and radius of (x−3)²+(y+4)²=9. Candidate answers: (A) Center (3,4); radius 3 (B) Center (3,−4); radius 3 (C) Center (3,−4); radius 9 (D) Center (−3,4); radius 3.

Learn the idea

Circle, parabola, ellipse and hyperbola equations encode geometric features through squared quantities and sign patterns. Standard forms locate centers, focal information, axes or vertices, with care to distinguish full lengths from half lengths. The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius. The constants inside squared differences locate the center rather than directly supplying axis intercepts.

Here is the method on the supplied teaching case. Compare to (x−h)²+(y−k)²=r². Here h=2, k=−3, r=√4=2. The signs inside the parentheses are opposite the center coordinates.

The signs in the differences are opposite the center’s coordinate signs. The right-side value is radius squared, so take its positive root rather than reporting that squared value as the radius.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Identify center and radius of (x−2)²+(y+3)²=4. Candidate answers: (A) Center (2,3); radius 2 (B) Center (2,−3); radius 2 (C) Center (2,−3); radius 4 (D) Center (−2,3); radius 2. Solution: Compare to (x−h)²+(y−k)²=r². Here h=2, k=−3, r=√4=2. The signs inside the parentheses are opposite the center coordinates. The final answer is Center (2,−3); radius 2. The signs in the differences are opposite the center’s coordinate signs. The right-side value is radius squared, so take its positive root rather than reporting that squared value as the radius.

Your turn

Check your reasoning

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Helpful tips
  • The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius.
  • The signs in the differences are opposite the center’s coordinate signs.
Identify center and radius of (x−3)²+(y+4)²=9.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

Center (3,−4); radius 3

Compare to (x−h)²+(y−k)²=r². Here h=3, k=−4, r=√9=3. The signs inside the parentheses are opposite the center coordinates.

Take it further

Changed-input task: Identify center and radius of (x−9)²+(y+10)²=81. Candidate answers: (A) Center (9,10); radius 9 (B) Center (9,−10); radius 81 (C) Center (9,−10); radius 9 (D) Center (−9,10); radius 9. Show the mathematical reason, not only the selected answer. Expected result: Center (9,−10); radius 9. Compare to (x−h)²+(y−k)²=r². Here h=9, k=−10, r=√81=9. The signs inside the parentheses are opposite the center coordinates.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Identify center and radius of (x−4)²+(y+5)²=16.
Need a hint?
  • The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius.
  • The signs in the differences are opposite the center’s coordinate signs.
2. Identify center and radius of (x−5)²+(y+6)²=25.
Need a hint?
  • The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius.
  • The signs in the differences are opposite the center’s coordinate signs.
3. Identify center and radius of (x−6)²+(y+7)²=36.
Need a hint?
  • The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius.
  • The signs in the differences are opposite the center’s coordinate signs.

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