Circles

Develop center-radius form, completing the square, coordinate distance.

Lesson 3 of 63 parts Practice Flashcards

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Center-Radius Form

Objective: Explain center-radius form, apply the rule, and justify relevant restrictions.

A circle with center (h,k) and radius r>0 satisfies (x−h)²+(y−k)²=r². The right side is squared radius, not radius itself. Radius is a nonnegative distance. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(x−h)²+(y−k)²=r²$

Worked examples

Try itFor (x−3)²+(y−5)²=49, find the radius.Show answer

Step 1: Compare with (x−h)²+(y−k)²=r². Step 2: Here r²=49. Step 3: Take the positive root: r=7.

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Center-Radius Form

Objective: Explain center-radius form, apply the rule, and justify relevant restrictions.

Formulas

$(x−h)²+(y−k)²=r²$

Worked examples

Try itFor (x−3)²+(y−5)²=49, find the radius.Show answer

Step 1: Compare with (x−h)²+(y−k)²=r². Step 2: Here r²=49. Step 3: Take the positive root: r=7.

Completing the Square

Objective: Explain completing the square, apply the rule, and justify relevant restrictions.

Group x and y terms, complete each square, and move constants to expose the center-radius form; keep both equation sides balanced. Adding to only one side changes the circle. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$x²−2hx+h²=(x−h)²$

Worked examples

Try itComplete the square in x²−6x+y²−10y+(-15)=0. What is the radius?Show answer

Step 1: Add 9 and 25 to form (x−3)² and (y−5)². Step 2: The equation becomes (x−3)²+(y−5)²=49. Step 3: The radius is √49=7.

Coordinate Distance

Objective: Explain coordinate distance, apply the rule, and justify relevant restrictions.

The distance formula comes from the Pythagorean theorem; points on a circle are all r units from center. Interpret coordinate units consistently. No, a distance cannot be negative. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$d=√((x₂−x₁)²+(y₂−y₁)²)$

Worked examples

Try itFind the distance from (3,5) to (6,9).Show answer

Step 1: The horizontal change is 3; vertical change is 4. Step 2: Apply √(3²+4²)=√25. Step 3: The distance is 5.

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