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Desmos: Circle Equations.
If the problem says the equation of a circle, type the circle equation into Desmos, EXACTLY the way it is written in the problem.
Read the finished equation
(x − h)² + (y − k)² = r²
(x − 5)² + (y + 1)² = 9
The center is the opposite of the numbers inside.
The center is (5, −1).
The radius is .
The radius is 3.
It still represents a circle
The equation represents a circle.
The x² and y² terms may be mixed with other terms.
Move those terms together, then complete the square.
Example
The equation represents a circle.
9 = 4x − y² − 6y − x²
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² − 4x + y² + 6y = −9
Complete the square.
Half of −4 is −2, and (−2)² = 4.
Half of 6 is 3, and 3² = 9.
Add 4 and 9 to both sides.
x² − 4x + 4 + y² + 6y + 9 = −9 + 4 + 9
−9 + 4 + 9 = 4
(x − 2)² + (y + 3)² = 4
The center is the opposite of the numbers inside.
The center is (2, −3).
The radius is .
The radius is 2.
Answer: center (2, −3), radius 2
Together
Work these three in a session. Type the circle equation into Desmos exactly the way it is written.
The equation represents a circle. Which choice is the center and the radius?
20 = 10y − y² − 4x − x²
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² + 4x + y² − 10y = −20
Complete the square.
Half of 4 is 2, and 2² = 4.
Half of −10 is −5, and (−5)² = 25.
Add 4 and 25 to both sides.
x² + 4x + 4 + y² − 10y + 25 = −20 + 4 + 25
−20 + 4 + 25 = 9
(x + 2)² + (y − 5)² = 9
The center is the opposite of the numbers inside.
The center is (−2, 5).
The radius is .
The radius is 3.
Answer: B. center (−2, 5), radius 3
The equation represents a circle. Which choice is the center and the radius?
21 + x² + y² + 6x + 8y = 0
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² + 6x + y² + 8y = −21
Complete the square.
Half of 6 is 3, and 3² = 9.
Half of 8 is 4, and 4² = 16.
Add 9 and 16 to both sides.
x² + 6x + 9 + y² + 8y + 16 = −21 + 9 + 16
−21 + 9 + 16 = 4
(x + 3)² + (y + 4)² = 4
The center is the opposite of the numbers inside.
The center is (−3, −4).
The radius is .
The radius is 2.
Answer: C. center (−3, −4), radius 2
The equation represents a circle. Which choice is the center and the radius?
4 = 12x − y² − 4y − x²
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² − 12x + y² + 4y = −4
Complete the square.
Half of −12 is −6, and (−6)² = 36.
Half of 4 is 2, and 2² = 4.
Add 36 and 4 to both sides.
x² − 12x + 36 + y² + 4y + 4 = −4 + 36 + 4
−4 + 36 + 4 = 36
(x − 6)² + (y + 2)² = 36
The center is the opposite of the numbers inside.
The center is (6, −2).
The radius is .
The radius is 6.
Answer: D. center (6, −2), radius 6
Independent Practice
Try these five on your own. Type the circle equation into Desmos exactly the way it is written.
The equation represents a circle. Which choice is the center and the radius?
10 + x² + y² − 2x − 10y = 0
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² − 2x + y² − 10y = −10
Complete the square.
Half of −2 is −1, and (−1)² = 1.
Half of −10 is −5, and (−5)² = 25.
Add 1 and 25 to both sides.
x² − 2x + 1 + y² − 10y + 25 = −10 + 1 + 25
−10 + 1 + 25 = 16
(x − 1)² + (y − 5)² = 16
The center is the opposite of the numbers inside.
The center is (1, 5).
The radius is .
The radius is 4.
Answer: A. center (1, 5), radius 4
The equation represents a circle. Which choice is the center and the radius?
9 = 6y − y² − 10x − x²
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² + 10x + y² − 6y = −9
Complete the square.
Half of 10 is 5, and 5² = 25.
Half of −6 is −3, and (−3)² = 9.
Add 25 and 9 to both sides.
x² + 10x + 25 + y² − 6y + 9 = −9 + 25 + 9
−9 + 25 + 9 = 25
(x + 5)² + (y − 3)² = 25
The center is the opposite of the numbers inside.
The center is (−5, 3).
The radius is .
The radius is 5.
Answer: B. center (−5, 3), radius 5
The equation represents a circle. Which choice is the center and the radius?
28 + x² + y² − 14x − 4y = 0
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² − 14x + y² − 4y = −28
Complete the square.
Half of −14 is −7, and (−7)² = 49.
Half of −4 is −2, and (−2)² = 4.
Add 49 and 4 to both sides.
x² − 14x + 49 + y² − 4y + 4 = −28 + 49 + 4
−28 + 49 + 4 = 25
(x − 7)² + (y − 2)² = 25
The center is the opposite of the numbers inside.
The center is (7, 2).
The radius is .
The radius is 5.
Answer: C. center (7, 2), radius 5
The equation represents a circle. Which choice is the center and the radius?
21 = 12y − y² − 2x − x²
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² + 2x + y² − 12y = −21
Complete the square.
Half of 2 is 1, and 1² = 1.
Half of −12 is −6, and (−6)² = 36.
Add 1 and 36 to both sides.
x² + 2x + 1 + y² − 12y + 36 = −21 + 1 + 36
−21 + 1 + 36 = 16
(x + 1)² + (y − 6)² = 16
The center is the opposite of the numbers inside.
The center is (−1, 6).
The radius is .
The radius is 4.
Answer: D. center (−1, 6), radius 4
The equation represents a circle. Which choice is the center and the radius?
33 + x² + y² − 6x + 10y = 0
Show Answer
Type the circle equation into Desmos, EXACTLY the way it is written in the problem.
(x − h)² + (y − k)² = r²
Move the x terms together and the y terms together.
x² − 6x + y² + 10y = −33
Complete the square.
Half of −6 is −3, and (−3)² = 9.
Half of 10 is 5, and 5² = 25.
Add 9 and 25 to both sides.
x² − 6x + 9 + y² + 10y + 25 = −33 + 9 + 25
−33 + 9 + 25 = 1
(x − 3)² + (y + 5)² = 1
The center is the opposite of the numbers inside.
The center is (3, −5).
The radius is .
The radius is 1.
Answer: B. center (3, −5), radius 1
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