Problem 32 - Precalculus: Functions and Graphs

Question

Find the values of \(b\) such that the system $$\left\\{\begin{aligned}x^{2}+y^{2} &=4 \\\y &=x+b\end{aligned}\right.$$ has (a) one solution (b) two solutions (c) no solution Interpret (a)- (c) graphically.

Step-by-Step Solution

Answer

(a) \(b = \pm 2\sqrt{2}\), (b) \(-2\sqrt{2} < b < 2\sqrt{2}\), (c) \(|b| > 2\sqrt{2}\).

Step 1: Understand the Problem

We need to find the values of \(b\) so that the system of equations results in one solution, two solutions, or no solutions. The system combines a circle equation, \(x^2 + y^2 = 4\), and a line equation, \(y = x + b\). The solutions represent the intersection points of these graphs.

Step 2: Solve for Intersection Points

To find the intersection points, substitute the equation of the line into the circle equation. Replace \(y\) in \(x^2 + y^2 = 4\) with \(x + b\), giving us \(x^2 + (x + b)^2 = 4\). Simplify to obtain a quadratic equation: \[2x^2 + 2bx + b^2 = 4\].

Step 3: Simplify the Quadratic Equation

Re-arrange the quadratic equation to standard form: \[2x^2 + 2bx + (b^2 - 4) = 0\]. This is a quadratic in \(x\).

Step 4: Determine the Discriminant

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(\Delta = b^2 - 4ac\) determines the number of solutions. Identify \(a = 2\), \(b = 2b\), and \(c = b^2 - 4\). Thus, the discriminant becomes: \[(2b)^2 - 4 \times 2 \times (b^2 - 4) = 4b^2 - 8(b^2 - 4)\].

Step 5: Simplify the Discriminant

Now simplify the expression: \[4b^2 - 8b^2 + 32 = -4b^2 + 32\]. The number of solutions are determined by the sign of this expression.

Step 6: Analyze the Discriminant and Solutions

- **One solution**: Occurs when the discriminant \(( -4b^2 + 32 )\) equals 0. Solve \(-4b^2 + 32 = 0\), leading to \(b^2 = 8\) or \(b = \pm\sqrt{8}\). Thus, \(b = \pm 2\sqrt{2}\).- **Two solutions**: Occurs when the discriminant is positive \((-4b^2 + 32 > 0)\). Solve for \(b^2 < 8\), meaning \(-2\sqrt{2} < b < 2\sqrt{2}\).- **No solution**: Occurs when the discriminant is negative \((-4b^2 + 32 < 0)\). Solve for \(b^2 > 8\), any \(b\) outside the range \(-2\sqrt{2}, 2\sqrt{2}\).

Step 7: Graphical Interpretation

Graphically, each case describes the relation between the circle and line: - **One solution**: The line is tangent to the circle at one point. - **Two solutions**: The line intersects the circle at two distinct points. - **No solution**: The line does not intersect the circle at any point.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of degree two. It follows the standard format:

• \(ax^2 + bx + c = 0\)

where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The solutions to this equation are called the roots of the quadratic and can be real or complex depending on the values of these constants. Quadratic equations graph as parabolas when plotted on a coordinate plane. The vertex of the parabola provides the maximum or minimum point. Additionally, parabolas are symmetric about the vertical line passing through this vertex.
In this exercise, after substituting the line equation into the circle equation, we end up with a quadratic equation in terms of \(x\). The specific form, \[2x^2 + 2bx + (b^2 - 4) = 0\], helps determine the intersections of the line with the circle by analyzing its roots.

Discriminant

The discriminant of a quadratic equation is a calculation that tells us about the nature and number of roots of the equation. For a quadratic equation in the form \(ax^2 + bx + c = 0\), calculate the discriminant \(\Delta\) as:

• \(b^2 - 4ac\)

The sign of the discriminant provides crucial information:

• If \(\Delta > 0\), the quadratic has two distinct real solutions.
• If \(\Delta = 0\), the quadratic has exactly one real solution (a repeated root).
• If \(\Delta < 0\), there are no real solutions, only complex solutions.

In relation to the given exercise, the discriminant was simplified to \[-4b^2 + 32\]. By analyzing its sign, you can determine when the line intersects the circle in one, two, or no points. For one solution, the discriminant equals zero; for two solutions, it’s positive; and when it's negative, there are no solutions.

Graphical Interpretation

Graphical interpretation is a powerful tool in understanding mathematical concepts by visualizing equations on a graph. For systems involving a circle and a line, the graph reveals their intersection in geometrical terms. Each scenario of intersections can visually indicate a different mathematical condition.

• One solution: The graph shows the line touching the circle exactly once, indicating it’s tangent to the circle.
• Two solutions: The line cross-cuts the circle at two distinct points, showing distinct intersection points.
• No solution: The line either lies entirely outside the circle or inside without touching, illustrating no intersection points.

This interpretation is invaluable for understanding the nature of solutions without detailed calculations.

Circle and Line Intersection

Understanding the intersection between a circle and a line is integral to solving systems of equations graphically or algebraically. A circle typically has the equation:

• \(x^2 + y^2 = r^2\), where \(r\) is the radius.

A line can be expressed as:

• \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

In this exercise, the circle equation is \(x^2 + y^2 = 4\), and the line is represented by \(y = x + b\).
There are three possibilities for the intersection:

• If one solution exists, the line "kisses" the circle at a single point (tangent).
• For two solutions, the line pierces through the circle creating two crossing points.
• If there are no solutions, the line avoids the circle entirely.

Analyzing the intersection points helps determine the variable \(b\) that leads to each solution type, making use of both algebraic and geometric insights.