Problem 61 - College Algebra

Question

Which one of the following is true? a. A system of two equations in two variables whose graphs represent a circle and a line can have four real solutions. b. A system of two equations in two variables whose graphs represent a parabola and a circle can have four real solutions. c. A system of two equations in two variables whose graphs represent two circles must have at least two real solutions. d. A system of two equations in two variables whose graphs represent a parabola and a circle cannot have only one real solution.

Step-by-Step Solution

Answer

Statement d is the only true statement.

Step 1: Analyzing Statement a

A system of two equations in two variables whose graphs represent a circle and a line can have at most two real solutions. This is because a line can intersect a circle at a maximum of two distinct points. Therefore, statement a is false.

Step 2: Analyzing Statement b

A system of two equations in two variables whose graphs represent a parabola and a circle can also have at most two real solutions. The reasoning is the same as in Step 1 - a parabola can intersect a circle at a maximum of two distinct points. Therefore, statement b is also false.

Step 3: Analyzing Statement c

A system of two equations in two variables whose graphs represent two circles could have one, two, or no real solutions, depending on their position relative to each other. Therefore, statement c, which suggests that there must be at least two real solutions, is also false.

Step 4: Analyzing Statement d

A system of two equations in two variables whose graphs represent a parabola and a circle indeed can have only one real solution. This would happen in the case where the parabola is tangent to the circle. Therefore, statement d is the only true statement.

Key Concepts

Circle and Line Intersection

When it comes to the intersection of a circle and a line in a system of equations, visualize a circle on the XY-plane. Now imagine a straight line passing through or touching that circle. Depending on how the line is positioned:

• The line might pass through the circle at two distinct points. In this case, there are exactly two real solutions.
• The line might be tangent to the circle, touching it at exactly one point. Here, we have one real solution.
• If the line doesn't touch the circle at all, there are no real solutions.

Therefore, the maximum number of real solutions for a circle and line system is two. Understanding this helps clarify why a statement suggesting up to four solutions would be incorrect.

Parabola and Circle Intersection

Consider the intersection between a parabola and a circle. A parabola has a distinct U-shape that can open up or down (or left or right in some contexts). When you overlay a parabola and a circle:

• They can intersect at two points, providing two real solutions.
• If the curve of the parabola just touches the edge of the circle, they are tangent, and there's one real solution.
• In cases where the parabola doesn't reach the circle at all, there are no real solutions.

Like the circle and line case, having four real solutions here is impossible. This is due to the symmetrical and curved nature of the shapes involved.

Real Solutions in Systems

Real solutions in systems refer to the actual points of intersection where two shapes cross each other in a coordinate plane. When solving systems involving shapes like lines, circles, and parabolas:

• Each point of intersection represents a set of real solutions.
• The nature and orientation of each shape affect how many times they intersect.
• Solving the system generally involves algebraic techniques to find these intersection points.

Real solutions tell us the number of simultaneous true results for both equations within a system.

Tangent Lines in Geometry

In geometry, a tangent line is a line that touches a curve at exactly one point—and doesn't cross it. This concept plays a crucial role in determining solutions within systems of equations:

• If a line is tangent to a circle, it means there is precisely one point of contact, offering one real solution.
• Similarly, if a parabola is tangent to a circle, it also results in one real solution.
• Tangency highlights a condition of exact intersection without crossing, differentiating it from a scenario of no solution (no intersection) or two solutions (two intersections).

Understanding tangent lines helps illuminate why certain geometric configurations result in only a single solution within systems of equations.