Circle Equations | Math Coach Amy

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What to Remember

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.

Then type the equation

Type a choice.

The right circle passes through the labeled point.

Its center is the labeled center.

Time saver

The center is the opposite of the numbers inside.

Center (4, −2) is written (x − 4)² + (y + 2)².

Cross out a choice whose signs give another center.

Example

What is the equation of a circle with a center of (4, −2) that contains the point (7, 2)?

A. (x − 4)² + (y + 2)² = 25

B. (x + 4)² + (y − 2)² = 25

C. (x − 4)² + (y + 2)² = 5

D. (x + 4)² + (y − 2)² = 5

Time Saver: the center is the opposite of the numbers inside.

The center (4, −2) is written (x − 4)² + (y + 2)².

Cross out B and D.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (4, −2) and (7, 2) labeled, and the equations (x − 4)² + (y + 2)² = 25 and (x − 4)² + (y + 2)² = 5.

The graph of (x − 4)² + (y + 2)² = 25 passes through (7, 2).

Its center is (4, −2).

Only that circle passes through the point.

Answer: A. (x − 4)² + (y + 2)² = 25

Together

Work these three in a session. Type the circle equation into Desmos exactly the way it is written.

What is the equation of a circle with a center of (−3, 4) that contains the point (−1, 1)?

A. (x + 3)² + (y − 4)² = 5 B. (x + 3)² + (y − 4)² = 13 C. (x − 3)² + (y + 4)² = 13 D. (x − 3)² + (y + 4)² = 5

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (−3, 4) is written (x + 3)² + (y − 4)².

Cross out C and D.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (−3, 4) and (−1, 1) labeled, and the equations (x + 3)² + (y − 4)² = 5 and (x + 3)² + (y − 4)² = 13.

The graph of (x + 3)² + (y − 4)² = 13 passes through (−1, 1).

Its center is (−3, 4).

Only that circle passes through the point.

Answer: B. (x + 3)² + (y − 4)² = 13

What is the equation of a circle with a center of (2, 5) that contains the point (5, 8)?

A. (x + 2)² + (y + 5)² = 18 B. (x + 2)² + (y + 5)² = 36 C. (x − 2)² + (y − 5)² = 18 D. (x − 2)² + (y − 5)² = 36

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (2, 5) is written (x − 2)² + (y − 5)².

Cross out A and B.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (2, 5) and (5, 8) labeled, and the equations (x − 2)² + (y − 5)² = 18 and (x − 2)² + (y − 5)² = 36.

The graph of (x − 2)² + (y − 5)² = 18 passes through (5, 8).

Its center is (2, 5).

Only that circle passes through the point.

Answer: C. (x − 2)² + (y − 5)² = 18

What is the equation of a circle with a center of (−5, −1) that contains the point (0, 1)?

A. (x + 5)² + (y + 1)² = 7 B. (x − 5)² + (y − 1)² = 7 C. (x − 5)² + (y − 1)² = 29 D. (x + 5)² + (y + 1)² = 29

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (−5, −1) is written (x + 5)² + (y + 1)².

Cross out B and C.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (−5, −1) and (0, 1) labeled, and the equations (x + 5)² + (y + 1)² = 7 and (x + 5)² + (y + 1)² = 29.

The graph of (x + 5)² + (y + 1)² = 29 passes through (0, 1).

Its center is (−5, −1).

Only that circle passes through the point.

Answer: D. (x + 5)² + (y + 1)² = 29

Independent

Independent Practice

Try these five on your own. Type the circle equation into Desmos exactly the way it is written.

What is the equation of a circle with a center of (3, −6) that contains the point (4, −1)?

A. (x − 3)² + (y + 6)² = 26 B. (x − 3)² + (y + 6)² = 6 C. (x + 3)² + (y − 6)² = 26 D. (x + 3)² + (y − 6)² = 6

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (3, −6) is written (x − 3)² + (y + 6)².

Cross out C and D.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (3, −6) and (4, −1) labeled, and the equations (x − 3)² + (y + 6)² = 26 and (x − 3)² + (y + 6)² = 6.

The graph of (x − 3)² + (y + 6)² = 26 passes through (4, −1).

Its center is (3, −6).

Only that circle passes through the point.

Answer: A. (x − 3)² + (y + 6)² = 26

What is the equation of a circle with a center of (−4, 2) that contains the point (−6, 6)?

A. (x − 4)² + (y + 2)² = 20 B. (x + 4)² + (y − 2)² = 20 C. (x − 4)² + (y − 2)² = 20 D. (x + 4)² + (y − 2)² = 4

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (−4, 2) is written (x + 4)² + (y − 2)².

Cross out A and C.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (−4, 2) and (−6, 6) labeled, and the equations (x + 4)² + (y − 2)² = 20 and (x + 4)² + (y − 2)² = 4.

The graph of (x + 4)² + (y − 2)² = 20 passes through (−6, 6).

Its center is (−4, 2).

Only that circle passes through the point.

Answer: B. (x + 4)² + (y − 2)² = 20

What is the equation of a circle with a center of (1, 3) that contains the point (−1, 8)?

A. (x − 1)² + (y + 3)² = 29 B. (x + 1)² + (y + 3)² = 29 C. (x − 1)² + (y − 3)² = 29 D. (x − 1)² + (y − 3)² = 7

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (1, 3) is written (x − 1)² + (y − 3)².

Cross out A and B.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (1, 3) and (−1, 8) labeled, and the equations (x − 1)² + (y − 3)² = 29 and (x − 1)² + (y − 3)² = 7.

The graph of (x − 1)² + (y − 3)² = 29 passes through (−1, 8).

Its center is (1, 3).

Only that circle passes through the point.

Answer: C. (x − 1)² + (y − 3)² = 29

What is the equation of a circle with a center of (6, −4) that contains the point (2, −7)?

A. (x − 6)² + (y + 4)² = 5 B. (x + 6)² + (y + 4)² = 25 C. (x + 6)² + (y − 4)² = 25 D. (x − 6)² + (y + 4)² = 25

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (6, −4) is written (x − 6)² + (y + 4)².

Cross out B and C.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (6, −4) and (2, −7) labeled, and the equations (x − 6)² + (y + 4)² = 5 and (x − 6)² + (y + 4)² = 25.

The graph of (x − 6)² + (y + 4)² = 25 passes through (2, −7).

Its center is (6, −4).

Only that circle passes through the point.

Answer: D. (x − 6)² + (y + 4)² = 25

What is the equation of a circle with a center of (−1, −4) that contains the point (5, −6)?

A. (x − 1)² + (y − 4)² = 40 B. (x + 1)² + (y + 4)² = 40 C. (x + 1)² + (y − 4)² = 40 D. (x + 1)² + (y + 4)² = 64

Show Answer

Time Saver: the center is the opposite of the numbers inside.

The center (−1, −4) is written (x + 1)² + (y + 4)².

Cross out A and C.

Type the circle equation into Desmos, EXACTLY the way it is written in the problem.

Desmos graph with (−1, −4) and (5, −6) labeled, and the equations (x + 1)² + (y + 4)² = 40 and (x + 1)² + (y + 4)² = 64.

The graph of (x + 1)² + (y + 4)² = 40 passes through (5, −6).

Its center is (−1, −4).

Only that circle passes through the point.

Answer: B. (x + 1)² + (y + 4)² = 40

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