Problem 40 - Algebra and Trigonometry

Question

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Step-by-Step Solution

Answer

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

Step 1: Understand the parabola structure

A parabola with its vertex at the origin and focus on the negative y-axis opens downward. The general equation for a vertical parabola is \( x^2 = 4py \), where the vertex is \((0,0)\) and \( p \) is the distance from the vertex to the focus.

Step 2: Define the focal diameter

The focal diameter (or the latus rectum) of the parabola is twice the absolute value of \( p \). In this problem, the focal diameter is 8, so we have \( |4p| = 8 \).

Step 3: Solve for p

Since \( |4p| = 8 \), we can solve for \( p \) by dividing both sides by 4. This gives us \( |p| = 2 \). Because the focus is on the negative y-axis, \( p = -2 \).

Step 4: Write the equation of the parabola

Substitute \( p = -2 \) into the general equation \( x^2 = 4py \). This gives us \( x^2 = 4(-2)y \) or \( x^2 = -8y \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a special point that provides crucial information about the shape and direction of a parabolic curve. In simpler terms, think of the vertex as the "tip" or "turning point" of the parabola. When a parabola is graphed, this is the point where it either reaches its maximum or minimum value. For ever more simplicity, imagine it as the bottom of a U-shaped curve—if it opens upwards—or the top of an upside-down U if it opens downwards.

In many equations, especially where the vertex is located at the origin, you’ll see the equation structured as either \( x^2 = 4py \) or \( y^2 = 4px \). When the vertex is at the origin,

• It’s conveniently located at the point (0,0) in the coordinate plane.
• This makes calculations and plotting much simpler.

The vertex acts as the starting point from which the parabola extends. In standard problems, once you identify the vertex, you can determine how the rest of the parabola behaves.

Focus and Directrix

To understand the focus and the directrix, envision a parabola as a curve where each point maintains an equal distance from a special point and a line. These are known as the "focus" and "directrix," respectively. The focus is a point inside the parabola, while the directrix is a line outside the parabola.

The relationship between the focus and the directrix is what gives the parabola its unique shape:

• The focus provides the direction in which the parabola opens.
• In our example, with the focus on the negative y-axis, the parabola opens downwards.
• The distance from any point on the parabola to the focus is the same as to the directrix.

This balance of distance makes the parabola symmetric, and knowing where the focus is can help plot the parabola accurately.

In the general equation, via the parameter \( p \), you can determine both the focus and the length of the distances from the vertex to these critical features. With the exercise's focus at \( (0,-p) \), you can see exactly where the downward direction of this particular parabola comes from.

Focal Diameter

The focal diameter, also known as the latus rectum, of a parabola is an important measure. It helps to define the "width" of the parabola along its axis of symmetry. The focal diameter is given by the formula of \( 4|p| \), where \( p \) is a fundamental value linked to the parabola's focus.

In this exercise, the focal diameter is specified as 8. By solving \( |4p| = 8 \), we discovered that \( |p| = 2 \). This is crucial because knowing \( p \) allows us to:

• Determine exactly where the focus is. Here it is at \( (0, -2) \).
• Understand the span of the parabola's "width" directly beneath the focus.

The focal diameter literally represents the length of a line segment that passes through the focus and is perpendicular to the axis of symmetry. This feature gives us a sense of how "open" or "closed" the parabolic curve will appear.

In summary, the focal diameter connects the width and position of the parabola, further emphasizing the balance between the opening size and the curve's steepness.