Determine The Relative Position Of The Line Y = 2x + 3 With Respect To The Parabola F(x) = X^2 + 4x + 12.

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In this article, we will explore how to determine the relative position of a line and a parabola. This is a fundamental concept in analytic geometry, with applications in various fields, including physics, engineering, and computer graphics. We will focus on a specific example: determining the relative position of the line d with the equation y = 2x + 3 and the parabola f(x) = x² + 4x + 12. Understanding the relationship between these two geometric figures involves finding their points of intersection, if any, and analyzing the discriminant of the resulting quadratic equation. This article will provide a step-by-step guide to solving this problem, ensuring clarity and a comprehensive understanding of the underlying principles.

Understanding the Concepts

Before diving into the calculations, let's clarify the possible relative positions of a line and a parabola. A line can interact with a parabola in three distinct ways:

  1. The line intersects the parabola at two distinct points: This occurs when the line crosses the parabola, creating two points of intersection. In this case, the quadratic equation formed by equating the line and parabola equations will have two distinct real roots.
  2. The line is tangent to the parabola: This happens when the line touches the parabola at exactly one point. The line grazes the parabola without crossing it. The quadratic equation will have exactly one real root (a repeated root).
  3. The line does not intersect the parabola: In this scenario, the line and parabola do not share any common points. The quadratic equation will have no real roots, indicating that the line and parabola do not intersect in the real plane.

To determine which of these scenarios applies to our specific problem, we will follow a systematic approach. The first step involves setting the equations of the line and parabola equal to each other. This will create a new equation whose solutions (if they exist) represent the x-coordinates of the intersection points. Next, we will rearrange this equation into a standard quadratic form (ax² + bx + c = 0). This form is crucial because it allows us to use the discriminant, a key tool for determining the nature of the roots.

The discriminant, denoted by Δ (delta), is calculated using the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant provides valuable information about the roots of the equation:

  • If Δ > 0, the equation has two distinct real roots.
  • If Δ = 0, the equation has one real root (a repeated root).
  • If Δ < 0, the equation has no real roots.

By analyzing the discriminant, we can determine whether the line intersects the parabola at two points, is tangent to the parabola, or does not intersect it at all. This method offers a powerful and efficient way to understand the geometric relationship between these two curves. This entire process is rooted in the principles of algebra and analytic geometry, making it a classic example of how these mathematical disciplines intertwine to solve geometric problems.

Step-by-Step Solution

Now, let's apply these concepts to our specific problem. We have the line d with the equation y = 2x + 3 and the parabola f(x) = x² + 4x + 12. Our goal is to determine their relative position.

1. Equate the Equations

To find the points of intersection, we set the equation of the line equal to the equation of the parabola:

2x + 3 = x² + 4x + 12

This equation represents the x-values where the line and parabola have the same y-value, meaning they intersect. Equating the equations is the crucial first step in finding the intersection points, as it translates the geometric problem into an algebraic one. This allows us to use algebraic tools to solve for the unknown x-values.

2. Rearrange into Quadratic Form

Next, we rearrange the equation into the standard quadratic form ax² + bx + c = 0. Subtracting 2x and 3 from both sides, we get:

0 = x² + 4x + 12 - 2x - 3

Simplifying the equation, we obtain:

x² + 2x + 9 = 0

Now we have a quadratic equation in the standard form, where a = 1, b = 2, and c = 9. Rearranging the equation into this form is essential because it allows us to easily identify the coefficients a, b, and c, which are necessary for calculating the discriminant. This step transforms the problem into a format that can be readily analyzed using the discriminant formula.

3. Calculate the Discriminant

The discriminant, Δ, is calculated using the formula Δ = b² - 4ac. Substituting our values, we get:

Δ = (2)² - 4 * 1 * 9

Δ = 4 - 36

Δ = -32

We find that the discriminant is Δ = -32. Calculating the discriminant is the pivotal step in determining the nature of the roots of the quadratic equation. The discriminant acts as a key indicator, revealing whether the equation has real roots, and if so, how many. This single value provides crucial information about the intersection of the line and the parabola.

4. Interpret the Discriminant

Since the discriminant Δ = -32 is negative, the quadratic equation x² + 2x + 9 = 0 has no real roots. This means there are no real values of x that satisfy the equation, indicating that the line and parabola do not intersect. The interpretation of the discriminant is the final step in determining the relative position of the line and the parabola. A negative discriminant definitively indicates that the line does not intersect the parabola in the real plane. This completes our analysis, providing a clear understanding of the geometric relationship between the two curves.

Conclusion

In conclusion, the discriminant, Δ = -32, is negative, which indicates that the line y = 2x + 3 does not intersect the parabola f(x) = x² + 4x + 12. This means the line and parabola do not share any common points. Understanding this concept is crucial in analytic geometry as it provides a method to determine the relationship between lines and parabolas without necessarily graphing them. The steps involved – equating the equations, rearranging into quadratic form, calculating the discriminant, and interpreting its value – provide a systematic approach to solving similar problems.

This method can be applied to any line and parabola, making it a versatile tool in mathematics. Furthermore, understanding the discriminant and its implications extends beyond just lines and parabolas. It is a fundamental concept in quadratic equations and is used extensively in various mathematical and scientific applications. Mastering this concept not only helps in solving geometric problems but also builds a strong foundation for advanced mathematical studies. The relationship between the discriminant and the nature of the roots is a powerful connection between algebra and geometry, showcasing the interconnectedness of these mathematical fields. By grasping these principles, students and enthusiasts can confidently tackle a wide range of problems involving quadratic equations and their geometric interpretations.

In summary, the analysis of the discriminant provides a clear and concise way to determine the relative position of a line and a parabola. This method is not only efficient but also offers valuable insights into the nature of quadratic equations and their geometric representations. The problem we solved serves as a great example of how algebraic tools can be used to understand and analyze geometric relationships, highlighting the beauty and power of mathematics.