Math / Geometry

QuestionDetermine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are a circle and a line can have four real ordered-pair solutions.
Choose the correct answer below. A. The statement is false. The correct statement is "A system of two equations in two variables whose graphs are a circle and a line can have two or one real ordered-pair solutions." B. The statement is false. The correct statement is "A system of two equations in two variables whose graphs are a circle and a line can have only three real ordered-pair solutions." C. The statement is false. The correct statement is "A system of two equations in two variables whose graphs are a circle and a line can have only one real ordered-pair solution." D. The statement is true.

Studdy Solution

STEP 1

1. The system consists of a circle and a line in a two-dimensional plane.
2. We are looking for the number of real ordered-pair solutions, which are the intersection points of the circle and the line.

STEP 2

1. Understand the possible intersections of a circle and a line.
2. Determine the maximum number of intersection points.
3. Evaluate the truth of the given statement.
4. Choose the correct answer.

STEP 3

Understand the possible intersections of a circle and a line:
- A line can intersect a circle at most at two points. - The line can be tangent to the circle, intersecting at exactly one point. - The line can also not intersect the circle at all, resulting in zero intersection points.

STEP 4

Determine the maximum number of intersection points:
- The maximum number of real ordered-pair solutions (intersection points) is 2.

STEP 5

Evaluate the truth of the given statement:
- The statement claims that there can be four real ordered-pair solutions. - Since the maximum number of intersection points is 2, the statement is false.

STEP 6

Choose the correct answer:
- The correct statement should reflect that the system can have 0, 1, or 2 real ordered-pair solutions. - Therefore, the correct answer is: A. The statement is false. The correct statement is "A system of two equations in two variables whose graphs are a circle and a line can have two or one real ordered-pair solutions."
The correct answer is:
$\boxed{A}$

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Studdy Solution

STEP 1

1. The system consists of a circle and a line in a two-dimensional plane.
2. We are looking for the number of real ordered-pair solutions, which are the intersection points of the circle and the line.

STEP 2

1. Understand the possible intersections of a circle and a line.
2. Determine the maximum number of intersection points.
3. Evaluate the truth of the given statement.
4. Choose the correct answer.

STEP 3

Understand the possible intersections of a circle and a line:
- A line can intersect a circle at most at two points. - The line can be tangent to the circle, intersecting at exactly one point. - The line can also not intersect the circle at all, resulting in zero intersection points.

STEP 4

Determine the maximum number of intersection points:
- The maximum number of real ordered-pair solutions (intersection points) is 2.

STEP 5

Evaluate the truth of the given statement:
- The statement claims that there can be four real ordered-pair solutions. - Since the maximum number of intersection points is 2, the statement is false.

STEP 6

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Studdy Solution

STEP 1

1. The system consists of a circle and a line in a two-dimensional plane.2. We are looking for the number of real ordered-pair solutions, which are the intersection points of the circle and the line.

STEP 2

1. Understand the possible intersections of a circle and a line.2. Determine the maximum number of intersection points.3. Evaluate the truth of the given statement.4. Choose the correct answer.

STEP 3

Understand the possible intersections of a circle and a line:- A line can intersect a circle at most at two points. - The line can be tangent to the circle, intersecting at exactly one point. - The line can also not intersect the circle at all, resulting in zero intersection points.

STEP 4

Determine the maximum number of intersection points:- The maximum number of real ordered-pair solutions (intersection points) is 2.

STEP 5

Evaluate the truth of the given statement:- The statement claims that there can be four real ordered-pair solutions. - Since the maximum number of intersection points is 2, the statement is false.

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The system consists of a circle and a line in a two-dimensional plane.
2. We are looking for the number of real ordered-pair solutions, which are the intersection points of the circle and the line.

1. Understand the possible intersections of a circle and a line.
2. Determine the maximum number of intersection points.
3. Evaluate the truth of the given statement.
4. Choose the correct answer.

Understand the possible intersections of a circle and a line:
- A line can intersect a circle at most at two points. - The line can be tangent to the circle, intersecting at exactly one point. - The line can also not intersect the circle at all, resulting in zero intersection points.

Determine the maximum number of intersection points:
- The maximum number of real ordered-pair solutions (intersection points) is 2.

Evaluate the truth of the given statement:
- The statement claims that there can be four real ordered-pair solutions. - Since the maximum number of intersection points is 2, the statement is false.