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Chapter 13: Problem 39

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(-4,5) \text {, directrix } x=0 \quad y^{2}-10 y+8 x+41=0 $$

Short Answer

Expert verified
The equation of the parabola is \((y - 5)^2 = -8(x + 2)\).

Step by step solution

01

Understand the Given Information

The problem provides the focus of the parabola at \((-4,5)\) and the directrix as \(x = 0\). We are also given the equation \(y^2 - 10y + 8x + 41 = 0\), which we need to rearrange or transform into the standard form of a parabola.
02

Determine the Type of Parabola

Given the directrix \(x = 0\), which is a vertical line, and the focus \((-4,5)\), we are dealing with a parabola that opens sideways. Parabolas that open sideways have equations in the form \((y - k)^2 = 4p(x - h)\).
03

Calculate the Vertex

For a parabola with a horizontal axis, the vertex \((h, k)\) is the midpoint between the focus and a point on the directrix. Here, the midpoint between the focus at \((-4,5)\) and a point at \((0,5)\) on the directrix is \((-2,5)\), so the vertex is \((-2,5)\).
04

Determine the "p" Value

The distance between the focus and the vertex is \(p\). Since the focus is at \((-4,5)\) and the vertex at \((-2,5)\), \(p = -2\). (The negative sign indicates that the parabola opens to the left.)
05

Write the Equation of the Parabola

Using the vertex at \((-2,5)\) and \(p = -2\), substitute into the standard form equation: \((y - 5)^2 = 4(-2)(x + 2)\)which simplifies to \((y - 5)^2 = -8(x + 2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola
The vertex of a parabola is a significant point that acts as a peak or a valley of the parabola shape, depending on its orientation. In this case, it is the point
  • directly between the focus and the directrix
  • where the parabola changes direction
For the problem at hand, the vertex can be calculated as the midpoint between the given focus, even if the directrix is not directly a coordinate. The coordinates for the vertex are found by averaging the coordinates along the axis of symmetry between the focus and a point directly across from it on the directrix. Here, with a focus at either side or top and bottom of the current point, is midway between the focus determine the midpoint using averages.
Focus and Directrix
A parabola is uniquely determined by a fixed point known as the focus, and a line called the directrix. The focus is a point from which distances are measured in forming a parabolic curve, and the directrix is a line used for these measurements too.
  • The distance from any point on the parabola to the focus is equal to the distance from the same point to the directrix.
  • The directrix stretch across the parabola, providing a referral line for constructing the shape.
      • In the given problem, the focus is located at (-4, 5), while the directrix is represented by the line x = 0. Understanding the position of focus and directrix helps identify the parabola's orientation, which is crucial for forming its equation.
Standard Form of a Parabola
The standard form of a parabolic equation presents the structure through which properties like the vertex and axis of symmetry can be easily identified. This equation can have different forms depending on the parabola's orientation.
  • For a parabola opening sideways, the standard form is i n the focus to the vertex is determined based on this standard setup.
  • If the parabola opens upwards or downwards, the form becomes
      • Given the problem, we adopted the form ing sideways. It reflects the nature of the parabola wherein the squares term involves the y variable, depicting left-right opening.
Horizontal Axis of Symmetry
The axis of symmetry is a line that divides the parabola into two mirror-image halves and controls the axis around which the parabola balances. For parabolas that open horizontally, this line runs parallel to the x-axis.
  • The axis of symmetry occurs through the vertex, setting the middle of the structure.
  • This remark reflects how uniformly the shape stretches on either side, either to the left or right.
      • In our problem, the horizontal axis of symmetry is the line where all calculations align, Because the parabola opens in a direction perpendicular to this axis, comprehending the position and influence of this symmetry line helps anticipate the overall geometry of the parabola.

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Most popular questions from this chapter

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 2 x^{2}+12 x+11 y^{2}-88 y+172=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ (x-5)^{2}+(y-7)^{2}=25 $$

What is the graph of the equation \(x^{2}+y^{2}=-4\) ? \(\mathrm{Ex}-\) plain your answer.

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ (y-3)^{2}=8(x-1) $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}-12 x+24=0 $$
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