Problem 44 - Intermediate Algebra

Question

For the exercises below, see the Concept Check in this section. How many real solutions are possible for a system of equations whose graphs are an ellipse and a line? Draw diagrams to illustrate each possibility.

Step-by-Step Solution

Answer

The system can have 0, 1, or 2 real solutions.

Step 1: Identify Possible Intersections

To find the number of real solutions, we need to consider how a line can intersect an ellipse. There are three possibilities: the line does not intersect the ellipse at all, the line is tangent to the ellipse touching it at exactly one point, or the line intersects the ellipse at two distinct points.

Step 2: Case 1: No Intersection

In this case, the line is entirely outside the ellipse and does not touch it at any point. This scenario results in zero real solutions because there are no points common to both the line and the ellipse.

Step 3: Case 2: Tangent Intersection

Here, the line is tangent to the ellipse, meaning it touches the ellipse at exactly one point. This scenario produces exactly one real solution, as there is only one point where the line and the ellipse intersect.

Step 4: Case 3: Two Intersections

For this situation, the line crosses the ellipse at two different points. Therefore, there are two real solutions corresponding to these two intersection points.

Key Concepts

Real Solutions in Ellipse-Line Intersections

When discussing the intersection of a line and an ellipse, we are fundamentally exploring the concept of real solutions. The term 'real solutions' refers to the points where both the curve of the ellipse and the straight line meet in the coordinate plane. These points are crucial because they represent the values that satisfy both equations involved: the equation of the ellipse and the equation of the line.
Real solutions vary based on how a line interacts with an ellipse, and there can be:

• Zero real solutions, which occurs when the line does not touch the ellipse at all.
• One real solution, which happens when the line is tangent to the ellipse, touching it at exactly one point.
• Two real solutions, resulting when the line intersects the ellipse at two distinct points.

Understanding how many real solutions exist helps us predict the behavior of these intersections without graphing them each time.

Using Systems of Equations for Intersection

To find the real solutions for an ellipse and line intersection, we apply systems of equations. A system of equations consists of two or more equations with the same variables. Solving these systems helps find the common solutions, which are the intersections in this case.
For the ellipse and line:

• The ellipse is often given by an equation in the form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where \( (h, k) \) is the center of the ellipse.
• The line can be described by a linear equation, such as: \[ y = mx + c \] where \(m\) is the slope and \(c\) is the y-intercept.

By solving this system, we can substitute the line's equation into the ellipse's equation and solve for one variable, typically finding the intersection points as these variable values. These algebraic solutions provide insights into how and where the shapes intersect.

Graphical Solutions of Ellipse and Line Intersections

Graphical solutions offer a visual approach to understanding where an ellipse and a line intersect. When you graph both equations on the same plane, the real solutions are visible as the distinct points where the ellipse and the line overlap, if any.
There are three possible scenarios you may observe:

• **No intersection**: The line is positioned such that it does not cross or touch any part of the ellipse.
• **Tangent intersection**: The line just grazes the ellipse, creating one single point of contact.
• **Intersection at two points**: The line slices through the ellipse, creating two distinct points where their paths meet.

Drawing these graphs helps in intuitively understanding the number and nature of solutions, providing a clear contrast to the algebraic approach.

Tangent Intersection of a Line and Ellipse

A tangent intersection between a line and an ellipse is a special case where the line touches the ellipse at exactly one point. This scenario is significant because it reflects a precise balance between the two shapes.
At the point of tangency, the derivatives of both the ellipse's edge and the line will be equal, meaning they share the same slope at that contact point. This situation produces exactly one real solution in terms of intersection points.
Recognizing a tangent intersection is useful in various real-world applications where minimal contact is required, such as in engineering design and navigation systems. It allows both the understanding and prediction of physical interactions between curved and linear paths.