Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Parabola with focus \((2,5)\) and directrix \(x=10\)

Step-by-Step Solution

Answer

The equation is \((y - 5)^2 = -16(x - 6)\).

Step 1: Identify Key Elements

We begin by identifying the given elements of the parabola: the focus, which is at (2,5), and the directrix, which is the vertical line x = 10. Our goal is to derive the equation of the parabola with these elements.

Step 2: Determine Vertex

The vertex of the parabola lies halfway between the focus and the directrix. The x-coordinate of the vertex will be the average of the x-coordinates of the focus and the directrix. Thus, the vertex x-coordinate is \( \frac{2 + 10}{2} = 6 \), and the vertex is at (6, 5).

Step 3: Calculate Distance 'p'

The parameter 'p' is the directed distance from the vertex to either the focus or the directrix. Since the focus is to the left of the directrix, the parabola opens towards the focus, to the left. The distance is \( |6 - 2| = 4 \); since it opens to the left, \( p = -4 \).

Step 4: Formulate Parabolic Equation

Since the parabola is horizontal (and opens leftward), we use the formula for a horizontal parabola: \((y - k)^2 = 4p(x - h)\). Substitute \( h = 6\), \( k = 5\), and \( p = -4\), giving us: \((y - 5)^2 = -16(x - 6)\).

Step 5: Expand and Simplify

Expanding \((y - 5)^2 = -16(x - 6)\) gives us \( y^2 - 10y + 25 = -16x + 96\). Rearranging terms leads to the standard form of the parabola: \( -16x = y^2 - 10y - 71 \) or equivalently, \( 16x = -y^2 + 10y + 71 \).

Key Concepts

Focus and Directrix

The concepts of focus and directrix are essential to understanding parabolas. A parabola is the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. In the exercise, the focus is at the point \(2,5\), and the directrix is the vertical line \(x = 10\).

The focus acts as a guide to where the parabola opens, while the directrix provides a boundary that the parabola never touches. In this example, since the focus is to the left of the directrix, the parabola opens to the left. This relationship is crucial when forming the parabola's equation because it informs the orientation and the direction the parabola will open in relation to these two points.

Understanding the positions of the focus and directrix helps in visualizing how the parabolic curve is defined, giving insights into its curvature and position in a coordinate plane.

Vertex of a Parabola

The vertex of a parabola is a pivotal point which determines the maximum or minimum position on the curve, depending on its orientation. In simpler terms, it's the tip of the parabola.

To find the vertex of a parabola, you calculate the midpoint between the focus and the directrix:

• In the given example, with a focus at \(2,5\) and a directrix of \(x = 10\), the vertex x-coordinate is the average of the x-coordinates of the focus and directrix: \((2 + 10)/2 = 6\).
• The y-coordinate remains the same as the focus for a horizontally oriented parabola, which is 5.

This results in the vertex positioned at \(6, 5\).

The vertex essentially serves as the balancing point for the parabola, illustrating the symmetry and is crucial for forming the correct parabolic equation.

Distance 'p' in Parabolas

The distance 'p' plays a crucial role in determining the shape and orientation of a parabola. In a parabola, 'p' is defined as the distance from the vertex to the focus or the vertex to the directrix. It tells us how far the vertex is from these elements and dictates how "spread out" the parabola is.

For horizontal parabolas like the one in our exercise, \(p\) is calculated as follows:

• The distance between the vertex at \(6, 5\) and the focus at \(2, 5\) is \(|6 - 2| = 4\).
• However, because the parabola opens to the left (focus is to the left), this distance is negative, so \(p = -4\).

This concept of 'p' helps frame the parabolic equation, indicating whether it opens leftward or rightward and how wide the parabola appears on a graph.

Horizontal Parabola Formula

When dealing with horizontal parabolas, we use a specialized form of the parabola equation: \((y - k)^2 = 4p(x - h)\). This formula is distinct from the standard \(y = ax^2 + bx + c\) form used for vertically aligned parabolas.

Here's how this works in our example:

• \(h\) and \(k\) are the coordinates of the vertex (here \(6, 5\)).
• The parameter \(p\) is calculated as explained, \(-4\), indicating the direction the parabola opens.

Substituting these values into the formula, we get: \( (y - 5)^2 = -16(x - 6) \).

This equation describes a parabola that opens to the left, confirmed by a negative \(p\) value. Understanding this formula is fundamental as it allows one to derive the parabolic shape and direction effectively from any given focus and directrix.