Problem 32 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((1,0), \quad\) directrix \(x=-5\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = 24(x - 1)\).

Step 1: Identify the Orientation

Since the vertex of the parabola is at \((1,0)\) and the directrix is a vertical line \(x = -5\), the parabola opens to the right. This indicates it's a horizontal parabola.

Step 2: Determine the Standard Form

The standard form of a horizontal parabola is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex of the parabola. Here, \(h=1\) and \(k=0\).

Step 3: Calculate the Distance to the Directrix

The distance \(p\) from the vertex to the focus equals the distance from the vertex to the directrix. The vertex \((1,0)\) is \(6\) units away from the directrix \(x = -5\). Therefore, \(p = 6\).

Step 4: Write the Equation

Substitute \(h = 1\), \(k = 0\), and \(p = 6\) into the standard form equation: \((y - 0)^2 = 4 \times 6 \times (x - 1)\). Simplify to get the equation \(y^2 = 24(x - 1)\).

Key Concepts

Parabola Equation

A parabola is one of the basic conic sections found in mathematics, characterized by its unique symmetrical U-shape. The equation of a parabola takes different forms depending on its orientation. For parabolas that open to the left or right, as in this example, we use the standard equation

• \((y - k)^2 = 4p(x - h)\)

In this equation:

• \((h, k)\) represents the vertex of the parabola.
• \(p\) is the distance from the vertex to the focus, a crucial detail for constructing the parabola.

The main idea here is understanding how each component of the equation factors into the parabola's shape and direction.
The values of \(h\), \(k\), and \(p\) determine its position and orientation in the coordinate system. A positive value of \(p\) means the parabola extends to the right, while a negative \(p\) denoting a left-opening parabola. The concept becomes clearer by substituting these values appropriately.

Vertex and Directrix

The vertex and directrix are essential components in forming the parabola.
In this exercise, the vertex is given as

• \((1,0)\).

The vertex is the point from which the parabola appears to "spring." It is essentially the midpoint of the parabola's curve and serves as a key anchor in the parabola equation.
The directrix, on the other hand, is a fixed line used to define the parabola's "flattest" point in the geometric plane. Here, it is given by the vertical line

• \(x = -5\).

The directrix works together with the vertex to define the focus—a point equidistant from the vertex like the directrix.
It serves to help construct and visualize the parabolic shape, as it sets a reference line opposite the focus. For any point on the parabola, the distance to the focus equals the distance to the directrix.

Horizontal Parabola

A horizontal parabola is a specific orientation where the parabola opens sideways instead of up or down. In our case, the nature of the given vertex and directrix tells us the parabola opens to the right.
When the vertex

• \((1,0)\)

is combined accordingly with the directrix equation

• \(x = -5\)

we realize it is a perfect scenario for a horizontally oriented parabola.
The standard process involves determining key features:

• Direction: known from whether \(p\) is positive (right) or negative (left).
• Equation form: hinging on specific vertex-directrix distance.
• Length of \(p\): measuring distance from vertex to directrix, precisely 6 units here.

These insights into the horizontal orientation allow one to understand and predict its geometric depiction efficiently.