Problem 27 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(-8,0)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x = -\frac{1}{32}y^2\).

Step 1: Understand the Parabola Definition

A parabola with a vertex at the origin and focus at \((-8, 0)\) is a horizontal parabola because the focus has an x-coordinate other than zero, while the y-coordinate is zero.

Step 2: Use the Standard Form of the Parabola Equation

For a parabola with a vertex at the origin \((0, 0)\) and a horizontal orientation, the standard form is \(x = ay^2\).

Step 3: Determine the Distance to the Focus

For a horizontal parabola, the distance \(p\) between the vertex and the focus is the absolute value of the focus's x-coordinate, \(p = 8\) since the focus is \((-8,0)\). However, considering the direction to the focus, we use \(p = -8\).

Step 4: Relate Focus Distance to Coefficient

The equation in vertex form is \(x = \frac{1}{4p}y^2\). Substitute \(p = -8\) to find \(a = \frac{1}{4(-8)} = -\frac{1}{32}\).

Step 5: Write the Final Equation

Substitute \(a = -\frac{1}{32}\) into the vertex form equation: \(x = -\frac{1}{32}y^2\). This is the equation of the parabola with a focus at \((-8, 0)\) and vertex at the origin.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a handy way to express the quadratic equation of a parabola. For a parabola with a vertex at the origin \(0, 0\), this form simplifies things greatly.The vertex form for a parabola that opens horizontally is given by the equation:\[x = a(y - k)^2 + h\]Here, \((h, k)\) is the vertex of the parabola. With the vertex at the origin, \(h = 0\) and \(k = 0\), so the equation becomes:\[x = ay^2\]The parameter \(a\) determines the shape and direction of the parabola.- If \(a\) is positive, the parabola opens to the right.- If \(a\) is negative, it opens to the left.Understanding the vertex form helps in easily identifying key features of the parabola, such as its vertex and the direction it opens.

Focus and Directrix of a Parabola

The focus and directrix are essential geometric entities that help in defining a parabola.### Role of the FocusThe focus of a parabola is a fixed point on its interior. For a horizontal parabola with a vertex at the origin \(0,0\) and a focus at \((-8, 0)\), the parabola opens horizontally and curves around this focus.- The focus is used to determine the parabola's orientation and shape.### Role of the DirectrixOpposite to the focus, the directrix of a parabola is a line.- For a horizontal parabola, it is vertical. The vertex is exactly halfway between the focus and the directrix.- In our case, since the focus is \((-8, 0)\), the directrix is the vertical line \((x = 8)\).The parabola is defined in such a way that for any point on the parabola, the distance to the focus is equal to the distance to the directrix.

Understanding Horizontal Parabolas

Horizontal parabolas are a specific orientation of parabolas. Unlike the more common vertical parabolas that open up or down, these parabolas open left or right.### Characteristics- ** Vertex **: The starting point is typically at the origin \(0, 0\) which simplifies equations.- ** Direction **: The orientation (left or right) depends on the sign of \(a\) in the equation \(x = ay^2\).### Equation of Horizontal ParabolasFor a horizontal parabola with a vertex at the origin, the equation takes the form:- \(x = ay^2\)Where:- \(a = \text{positive}\) means the parabola opens to the right.- \(a = \text{negative}\) means it opens to the left, like in our exercise where \(a = -\frac{1}{32}\).Focusing on these key elements helps simplify the understanding of how horizontal parabolas behave, making it easier to graph and comprehend their properties.