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Magazine Chapter 8: Problem 38
Find the values of \(b\) such that the system has $$\left\\{\begin{aligned} x^{2}+y^{2} &=4 \\ y &=x+b\end{aligned}\right.$$ (a) one solution (b) two solutions (c) no solution Interpret (a)-(c) graphically.
Short Answer
Expert verified
(a) \(b = \pm\sqrt{2}\); (b) \(-2 < b < 2\) but \(b \neq \pm\sqrt{2}\); (c) \(|b| > 2\).
Step by step solution
01
Setting Up the Problem
We are given the system of equations \(x^2 + y^2 = 4\) and \(y = x + b\). The first equation represents a circle with radius 2 centered at the origin, while the second equation is a line with a slope of 1 and a y-intercept of \(b\). We need to find the values of \(b\) for different numbers of intersections between the line and the circle.
02
Substitute for y
Using the equation \(y = x + b\), substitute \(y\) into the circle's equation to obtain a single equation in terms of \(x\):\[x^2 + (x + b)^2 = 4\].
03
Expand and Simplify
Expand the equation \(x^2 + (x + b)^2 = 4\):\[x^2 + x^2 + 2xb + b^2 = 4\]which simplifies to\[2x^2 + 2xb + b^2 = 4\].
04
Rearranging into a Quadratic Form
Rearrange the equation to standard quadratic form:\[2x^2 + 2xb + (b^2 - 4) = 0\].Divide through by 2 to simplify:\[x^2 + xb + \frac{b^2 - 4}{2} = 0\].
05
Analyze the Discriminant
To determine the number of intersections, analyze the discriminant \(D\) of the quadratic equation \(x^2 + xb + \frac{b^2 - 4}{2} = 0\), given by:\[D = b^2 - 4 \times \frac{(b^2 - 4)}{2}\].
06
Solving for One Solution
For the equation to have one solution, \(D = 0\). Calculate the values of \(b\) that satisfy this as:\[b^2 = 2(b^2 - 4)\]Solve for \(b\), resulting in:\[b = \pm\sqrt{2}\].
07
Solving for Two Solutions
For two solutions, \(D > 0\). This implies:\[b^2 > 2(b^2 - 4)\].Upon simplification, check when \(D > 0\) using values outside \(+\sqrt{2}, -\sqrt{2}\) achieved by reflective values within permissible bounds.
08
Solving for No Solution
For no solution, \(D < 0\). This implies:\[b^2 < 2(b^2 - 4)\].Upon simplification, verify this does not hold true in any practical range as it involves a reflection outside achievable bounds.
09
Graphical Interpretation
Graphically, the circle is static with center at the origin, and the line moves vertically as \(b\) changes.- One solution: The line is tangent to the circle at two symmetrical points produced by the condition \( b = \pm\sqrt{2} \).- Two solutions: The line cuts the circle at two distinct points.- No solution: The line doesn't intersect the circle at all, which can occur if \(|b| > 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Discriminant
Let's talk about what a quadratic discriminant is and why it matters. The discriminant is a specific part of a quadratic equation's formula that helps us understand the nature of the equation's roots. When we have a quadratic equation in the form: \[ax^2 + bx + c = 0\] the discriminant \(D\) is given by the expression:\[D = b^2 - 4ac\].
Here's how the discriminant reveals the number of solutions:
Here's how the discriminant reveals the number of solutions:
- **One Solution**: If the discriminant \(D = 0\), then the quadratic equation has only one solution. This means the parabola touches the x-axis at one point, a tangent contact with another curve like in our circle and line problem.
- **Two Solutions**: If \(D > 0\), there are two distinct solutions, which means the parabola intersects the x-axis at two points.
- **No Solution**: If \(D < 0\), the equation has no real solutions. In this scenario, the parabola does not touch the x-axis at all.
System of Equations
A system of equations is when two or more equations work together to find a solution or set of solutions. In our exercise, we formed a system of equations with a circle \(x^2 + y^2 = 4\) and a line \(y = x + b\).
Here's what each part means for intersections:
Here's what each part means for intersections:
- **Circle Equation**: It represents all points on a plane that are equidistant from a fixed point, which is the center. Here, the circle has a center at the origin \((0,0)\) and a radius of 2.
- **Line Equation**: This equation describes a straight line, which is a collection of points in a plane that extends forever in both directions. The line has a slope of 1, rising and running equally, with a vertical shift determined by \(b\).
Circle and Line
Understanding the geometric relationship between a circle and a line is key to solving this exercise. Each figure has unique properties:
- **Circle**: A circle is defined by its center point and radius. All points on the circle are of equal distance from the center, making it perfectly symmetrical.
- **Line**: A line can cross a circle, touch it, or not meet it at all. The line's slope and position (dictated by \(b\) in \(y = x + b\)) affect its intersection with the circle.
- **One solution**: The line is tangent to the circle, meaning it just touches the circle at a single point. Visually, it means \(b\) is precisely adjusted to make the line touch the circle's edge.
- **Two solutions**: The line crosses the circle at two points. Depending on \(b\), it will enter and exit the circle at distinct coordinates.
- **No solution**: The line doesn't intersect the circle at all. In simple terms, \(b\) makes the line too far away to touch the circle.
