Question

WRITING Describe the possible solutions of a system that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justify your answers.

Short answer

A system that contains a quadratic equation and a circle can have either no solution, one solution or two solutions depending upon whether the parabola and circle don't intersect, touch at a point or intersect at two points respectively. Similarly, a system containing two circles can also have no solution, one solution or two solutions based on whether the circles are separate, tangent to each other or intersecting at two points.

Step-by-step solution

Understanding Quadratic and Circle Equations

Firstly, realize that a quadratic equation represents a parabola, while the equation of a circle represents, well a circle. The intersection points between these two elements represent the solutions of the system.

Solutions for Quadratic and Circle Equations

When a circle and a parabola intersect, there are three possible scenarios: (i) they don't intersect at all, (ii) they intersect at one point (tangent), or (iii) they intersect at two points. This corresponds to the idea that the quadratic equation (parabola) might not intersect, just touch or cross the circle.

Understanding Two Circles Intersection

When dealing with two circles, there are again three scenarios that may occur: (i) the circles do not intersect (they are separate), (ii) they intersect at only one point (they are tangent to each other), (iii) they intersect at two points.

Sketching the Graphs

In order to understand the solutions visually, sketch the graphs for each scenario on a Cartesian plane. This helps significantly in intuiting the possible solutions of the system.

Key concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation and is generally written in the form of \( ax^2 + bx + c = 0 \). The graph of a quadratic equation is a curve called a parabola. Parabolas can open upwards or downwards depending on the sign of \( a \).

When it comes to finding the roots of a quadratic equation, we are essentially looking for the values of \( x \) that make the equation equal zero. These roots can be real or complex and depend on the discriminant \( b^2 - 4ac \).

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution (the parabola touches the x-axis).
• If \( b^2 - 4ac < 0 \), there are no real solutions (the parabola does not intersect the x-axis), but there are two complex solutions.

These roots are crucial because they represent the points where the parabola intersects the x-axis, which is a fundamental aspect in graphically understanding system solutions.

Equation of a Circle

The equation of a circle in the Cartesian plane is given by \( (x - h)^2 + (y - k)^2 = r^2 \) where \( (h,k) \) is the center of the circle, and \( r \) is its radius. All points \( (x,y) \) that satisfy this equation lie exactly \( r \) units away from the center, hence forming the circular shape.

Understanding the circle's equation is vital when solving systems that include both a circle and another curve, such as quadratic equations. Depending on the position of the circle in relation to the parabola, different numbers of intersection points can be found. It is the locus of all those points that equidistance from a single point known as the center.

• If the distance between the circle's center and the vertex of the parabola is greater than the radius, there are no intersection points.
• If the distance is equal to the radius, the circle is tangent to the parabola at one point.
• If the distance is less than the radius, the circle and parabola intersect at two points.

Visually identifying these relationships can help in anticipating the number of solutions without extensive calculations.

Graphing Solutions

Graphing solutions of equations on a Cartesian plane allows us to visually interpret the possible intersections and points of tangency between curves. This graphical approach can often offer a more intuitive understanding of the system compared to algebraic solutions alone.

To graph a solution, you start by plotting the individual equations. For instance, you'd plot a parabola for the quadratic equation and a circle for its respective equation. By examining the graphs, you can identify the intersection points which represent the x and y values that solve both equations simultaneously.

Detailed Graphing

For proper graphing, consider the following aspects:

• Correctly identify and plot the vertex of the parabola and the center of the circle.
• Determine the direction in which the parabola opens (upward or downward).
• Mark the radii of the circle on the graph to ensure accuracy in size.

Graphing not only improves conceptual understanding but also visually reinforces the reason behind the number of solutions discussed algebraically.

Intersection of Curves

The intersection of curves, such as parabolas represented by quadratic equations and circles, is fundamental to solving systems of equations graphically. The points where the curves intersect are the solutions to the system. There may be zero, one, or two such points, depending on the relative position and shape of the curves.

When two curves intersect at a single point, they are tangent to each other at that point. If they intersect at two points, those points are where both curves pass through the same position in the plane. And if there's no intersection, then the curves do not share common points within the range of interest.

Tangible Intersection Scenarios

Here are possible scenarios for the intersection of two circles:

• If no intersection occurs, the circles are too far apart or one is entirely within the other without touching.
• One intersection point, or tangent, occurs when the circles just touch at one point.
• Two intersections points occur when the circles cross each other, sharing two points.

Understanding these intersection scenarios helps solve complex systems by visualizing and reasoning through the positions and shapes of the curves involved.