Problem 44 - Algebra 2

Question

CHALLENGE For Exercises \(44-48,\) find all values of \(k\) for which the system of equations has the given number of solutions. If no values of \(k\) meet the condition, write none. $$ x^{2}+y^{2}=k^{2} \quad \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$ no solutions

Step-by-Step Solution

Answer

The system has no solutions for any \( k < 2 \).

Step 1: Understand Geometric Representation

The first equation \( x^2 + y^2 = k^2 \) represents a circle centered at the origin with radius \( k \). The second equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) represents an ellipse centered at the origin with semi-major axis \(3\) (along the x-axis) and semi-minor axis \(2\) (along the y-axis).

Step 2: Consider the Circle and Ellipse Positions

For the system to have no solutions, the circle and the ellipse must not intersect at any point. Since both are centered at the origin, for no intersection, the circle's radius \( k \) must be less than \(2\), which is the semi-minor axis of the ellipse (the shortest distance from the center to the ellipse).

Step 3: Calculate the Condition for No Solutions

The circle and the ellipse do not intersect if and only if the radius of the circle \(k\) is less than \(2\). Therefore, the values of \(k\) for which the system has no solutions are values \(k < 2\).

Step 4: Conclusion on Values of k

Thus, the condition for the system of equations to have no solutions is that \(k\) must be a value less than \(2\). Any positive \(k < 2\) results in no intersection between the circle and the ellipse.

Key Concepts

Circle

A circle is a collection of points in a plane, all equidistant from a given point called the center. In the equation \( x^2 + y^2 = k^2 \), the center is at the origin \((0,0)\), and the radius is \(k\). This means all points \((x, y)\) that satisfy this equation are the same distance \(k\) from the center.

Key characteristics of circles include:

• The radius (\(k\)), which defines the distance from the center to any point on the circle.
• A continuous, curved path that is perfectly symmetrical.

Understanding the circle's geometric representation is important in problems involving systems of equations, as it describes the potential region of overlap with other shapes like ellipses.

Ellipse

Ellipses are smooth, symmetric curves that resemble stretched circles. The defining equation of an ellipse is a handy tool in algebra, and for this problem, the given ellipse is \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \). Here, the ellipse is centered at the origin and has two main axes: the semi-major axis of length 3 along the x-axis, and the semi-minor axis of length 2 along the y-axis.

Essential features of ellipses include:

• The semi-major axis, which is the longest radius that determines the ellipse's length.
• The semi-minor axis, which is the shortest radius that measures the ellipse's height.
• Ellipsoidal shape, offering a wide range of physical and graphical applications.

In systems of equations, identifying these axes helps determine intersections with other curves like circles.

Intersection

The intersection of two shapes refers to the points where they meet or overlap. In the solution process, determining intersection is key. When we have a circle and an ellipse centered at the same point, we consider their respective dimensions.

For no intersection in the given problem, the circle's radius \(k\) must be less than the ellipse's smallest width component, the semi-minor axis (2). This means:

• No points on the circle coincide with points on the ellipse.
• Ensuring a circle's radius does not overlap begins with comparing its radius to the dimensions of intersecting shapes like ellipses.

Using visual aids and radius comparisons can help conceptualize why certain solutions work, or in this case, don't work.

Solutions in Algebra

In algebra, finding solutions for a system of equations involves determining where equations intersect, if at all. For the given task, we analyzed when the circle \(x^2 + y^2 = k^2\) and the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) don't share any points.

This involves the idea of incompatible dimensions:

• The system has no solutions when \(k < 2\), as this ensures the circle is too small to reach any points on the ellipse.
• By altering parameters like \(k\), one can explore the conditions that affect solutions.

This concept teaches the importance of analyzing geometric and algebraic properties together, leading to comprehensive understanding in algebraic problem-solving.