Parabola Equation Directrix On Negative Y-Axis

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When delving into the fascinating world of conic sections, the parabola stands out as a fundamental shape with numerous applications in physics, engineering, and mathematics. A parabola is defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. Understanding the relationship between the vertex, focus, and directrix is crucial for determining the equation of a parabola.

In this article, we will explore parabolas with a vertex at the origin (0,0) and a directrix that intersects the negative part of the y-axis. This specific configuration provides valuable insights into the properties and equations of parabolas. We will dissect the characteristics of such parabolas, derive their equations, and analyze example problems to solidify our understanding. By the end of this discussion, you will be well-equipped to identify the equation of a parabola given its vertex and directrix orientation.

Key Concepts: Vertex, Focus, and Directrix

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Before diving into the specifics, let's revisit the core components of a parabola. The vertex is the point on the parabola that is closest to both the focus and the directrix. It serves as the turning point of the parabola. The focus is a fixed point inside the curve of the parabola, and the directrix is a fixed line outside the curve. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. This fundamental property defines the shape of the parabola.

For a parabola with its vertex at the origin (0,0), the orientation of the parabola is determined by the position of its focus and directrix. If the directrix is a horizontal line, the parabola opens either upwards or downwards. If the directrix is a vertical line, the parabola opens either to the right or to the left. The focus always lies on the axis of symmetry of the parabola, which is a line that passes through the vertex and divides the parabola into two symmetrical halves.

Parabolas Opening Upwards or Downwards

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Consider a parabola opening upwards or downwards with its vertex at (0,0). In this case, the axis of symmetry is the y-axis. The equation of such a parabola takes the form x^2 = 4py, where p is the distance from the vertex to the focus and also the distance from the vertex to the directrix. If p > 0, the parabola opens upwards, and the focus is at (0, p), while the directrix is the horizontal line y = -p. Conversely, if p < 0, the parabola opens downwards, the focus is at (0, p), and the directrix is the horizontal line y = -p.

Parabolas Opening to the Right or Left

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Now, let's consider a parabola opening to the right or left, again with its vertex at (0,0). Here, the axis of symmetry is the x-axis. The equation of this type of parabola is y^2 = 4px, where p retains its meaning as the distance from the vertex to the focus and to the directrix. If p > 0, the parabola opens to the right, the focus is at (p, 0), and the directrix is the vertical line x = -p. If p < 0, the parabola opens to the left, the focus is at (p, 0), and the directrix is the vertical line x = -p.

Analyzing the Given Condition: Directrix Crossing the Negative Y-Axis

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The core of the problem lies in the condition that the parabola has a directrix that crosses the negative part of the y-axis. This provides us with a crucial piece of information about the orientation and equation of the parabola. Since the directrix is a horizontal line intersecting the negative y-axis, we know that the directrix has the form y = -p, where -p is a negative value. This implies that p must be positive.

Furthermore, because the directrix is a horizontal line, the parabola must open either upwards or downwards. Since the directrix is below the vertex (0,0), the parabola must open upwards. This is because the parabola always curves away from the directrix and towards the focus. The focus, in this case, will lie above the vertex.

Given that the parabola opens upwards and has a vertex at (0,0), we know that its equation will be of the form x^2 = 4py, where p > 0. The value of p determines the “width” of the parabola; a larger p results in a wider parabola, while a smaller p results in a narrower parabola.

Implications for the Equation

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The fact that the directrix crosses the negative y-axis restricts the possible equations for the parabola. We've established that the equation must be of the form x^2 = 4py, where p > 0. This eliminates any equations where the coefficient of y is negative, as that would imply the parabola opens downwards. It also eliminates equations involving y^2, as those represent parabolas that open to the right or left.

Evaluating the Options

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Now, let's analyze the given options in light of our understanding:

  1. x^2 = -4y: This equation has a negative coefficient for y, which means p would be negative. This indicates a parabola opening downwards, which contradicts our condition that the directrix crosses the negative y-axis and thus the parabola opens upwards. Therefore, this option is incorrect.

  2. x^2 = 4y: This equation has a positive coefficient for y, implying p > 0. This fits our condition of a parabola opening upwards. The directrix for this parabola would be y = -1, which intersects the negative y-axis. Therefore, this option is a potential correct answer.

  3. y^2 = 4x: This equation represents a parabola that opens to the right, as it is in the form y^2 = 4px. This contradicts our condition that the directrix crosses the negative y-axis, which implies a parabola opening upwards or downwards. Therefore, this option is incorrect.

  4. y^2 = -4x: This equation represents a parabola that opens to the left, as it is in the form y^2 = 4px with a negative coefficient. This also contradicts our condition for the same reasons as the previous option. Therefore, this option is incorrect.

Conclusion

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Based on our analysis, the only equation that satisfies the condition of a parabola with a vertex at (0,0) and a directrix crossing the negative y-axis is x^2 = 4y. This equation represents a parabola opening upwards with a focus at (0,1) and a directrix at y = -1. Understanding the relationship between the vertex, focus, and directrix is paramount in determining the equation of a parabola.

By carefully considering the implications of the directrix's position, we were able to narrow down the possibilities and identify the correct equation. This exercise highlights the importance of a strong grasp of fundamental concepts in solving mathematical problems. Remember to always visualize the parabola and its components to aid in problem-solving.

Final Answer

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The correct equation is x^2 = 4y.