Problem 74 - Algebra and Trigonometry

Question

Determine whether each 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 parabola and a circle can have four real ordered-pair solutions.

Step-by-Step Solution

Answer

The statement is false. The correct statement is: A system of two equations in two variables whose graphs are a parabola and a circle can have at most two real ordered-pair solutions.

Step 1: Analyzing the shapes

A parabola is a curve where any point is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix). A circle is a shape consisting of all points in a plane that are a given distance from a given point (center). When these two shapes are plotted, we need to identify possible points of intersection.

Step 2: The intersection points

The statement is that there can be four real ordered-pair solutions, meaning, there can be four intersection points. However, imagining or sketching these graphs, it can be seen that a parabola and a circle can intersect at at most two points.

Step 3: Correcting the statement

Based on the analysis, the statement provided is false. So to make it a true statement, it should be changed to say: 'A system of two equations in two variables whose graphs are a parabola and a circle can have at most two real ordered-pair solutions.'