Problem 39 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \((y - k)^2 = 4p(x - h)\).

Step 3: 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 \((-2,5)\).

Step 4: 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.)

Step 5: 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 \((y - 5)^2 = -8(x + 2)\).

Key Concepts

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.