Parabola Intersection: Find The Value Of C

by Alex Johnson 43 views

Let's dive into a fascinating problem involving parabolas and lines! Our goal is to find the specific value of 'c' where the graph of the equation y = -x² + 9x - 100 intersects the horizontal line y = c at precisely one point. This means the line is tangent to the parabola.

Understanding the Problem: Tangency and Quadratic Equations

At its heart, this problem involves understanding what it means for a line to be tangent to a curve, specifically a parabola. When a line is tangent to a parabola, it touches the parabola at only one point. Algebraically, this translates to a quadratic equation having exactly one solution. We'll use this concept to solve for the value of 'c'. Let's break down the process step-by-step:

  1. Setting up the Equation: We begin by setting the equation of the parabola equal to the equation of the line: −x² + 9x − 100 = c. This represents the x-values where the parabola and the line intersect. The question specifies that there is exactly one intersection, so we need to find the value of c such that the equation has exactly one solution.

  2. Rearranging into Standard Quadratic Form: Now, let's rearrange this equation into the standard quadratic form, ax² + bx + c = 0. This makes it easier to apply the discriminant. Add x², subtract 9x, and add 100 to both sides, yielding: 0 = x² - 9x + (100 + c). Now the question is set in a format suitable for applying the discriminant condition for the existence of a unique solution.

  3. Applying the Discriminant: The discriminant, denoted as Δ, of a quadratic equation ax² + bx + c = 0 is given by the formula Δ = b² - 4ac. The discriminant tells us about the nature of the roots (solutions) of the quadratic equation:

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

    Since we want the line to intersect the parabola at exactly one point, we need the discriminant to be equal to zero. In our rearranged equation, a = 1, b = -9, and the constant term is (100 + c). Thus, we set up the equation:

    Δ = (-9)² - 4(1)(100 + c) = 0

  4. Solving for c: Now we solve for c:

    • 81 - 4(100 + c) = 0
    • 81 - 400 - 4c = 0
    • -319 - 4c = 0
    • -4c = 319
    • c = -319/4

Therefore, the value of c for which the line y = c intersects the parabola y = -x² + 9x - 100 at exactly one point is c = -319/4. This corresponds to option C.

Alternative Approach: Completing the Square

Another method to solve this problem involves completing the square. This approach gives a clearer geometric understanding of what's happening with the parabola.

  1. Rewrite the Parabola Equation: Start with the equation of the parabola: y = -x² + 9x - 100. We want to rewrite it in vertex form, which is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

  2. Complete the Square:

    • Factor out the negative sign from the x² and x terms: y = -(x² - 9x) - 100

    • To complete the square, we need to add and subtract (9/2)² = 81/4 inside the parenthesis:

      y = -(x² - 9x + 81/4 - 81/4) - 100 y = -((x - 9/2)² - 81/4) - 100 y = -(x - 9/2)² + 81/4 - 100

  3. Simplify to Vertex Form:

    • y = -(x - 9/2)² + 81/4 - 400/4
    • y = -(x - 9/2)² - 319/4

    Now the equation is in vertex form: y = -(x - 9/2)² - 319/4. The vertex of the parabola is at the point (9/2, -319/4).

  4. Find the Value of c: Since the parabola opens downward (due to the negative sign in front of the x² term), the vertex represents the maximum point on the parabola. For the line y = c to intersect the parabola at exactly one point, it must pass through the vertex. Therefore, c must be equal to the y-coordinate of the vertex:

    c = -319/4

This confirms our previous result, showing that the value of c is indeed -319/4.

Key Concepts Revisited

Let's solidify our understanding by revisiting the key concepts used in solving this problem:

  • Parabola: A parabola is a U-shaped curve defined by a quadratic equation. Its vertex represents either the maximum or minimum point on the curve.
  • Tangent Line: A tangent line touches a curve at only one point. In the context of a parabola and a horizontal line, tangency occurs at the vertex.
  • Discriminant: The discriminant of a quadratic equation determines the number and nature of its roots. A discriminant of zero indicates exactly one real root.
  • Completing the Square: A technique used to rewrite a quadratic expression in vertex form, revealing the vertex of the parabola.

Common Mistakes to Avoid

When tackling problems like this, be mindful of these common pitfalls:

  • Sign Errors: Pay close attention to signs, especially when completing the square or applying the discriminant formula. A small sign error can lead to a completely wrong answer.
  • Incorrectly Applying the Discriminant: Ensure you correctly identify the coefficients a, b, and c in the quadratic equation before using the discriminant.
  • Misunderstanding Tangency: Remember that tangency implies exactly one point of intersection, which corresponds to a quadratic equation having a single, repeated root.
  • Algebraic Errors: Always double-check your algebraic manipulations to avoid mistakes in rearranging equations or simplifying expressions.

Conclusion

In summary, we found the value of c for which the line y = c intersects the parabola y = -x² + 9x - 100 at exactly one point. We achieved this using both the discriminant method and the completing the square method, both leading to the solution c = -319/4. Understanding the relationship between quadratic equations, parabolas, and their properties allows us to solve such problems efficiently and accurately. Remember to practice and apply these concepts to similar problems to enhance your skills in algebra and analytic geometry. To further enhance your understanding of quadratic equations and parabolas, consider exploring resources like Khan Academy's Quadratic Equations Section.