Problem 32 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Focus \((1,-7),\) directrix \(x=-5\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y = \frac{6}{7}x - \frac{25}{14}\).

Step 1: Understand the Components

A parabola is defined as the set of all points that are equidistant from a point called the focus and a line called the directrix. Here, the focus is given as \((1,-7)\) and the directrix is \(x = -5\).

Step 2: Use the Definition of Parabola

To find the equation, we use the definition of a parabola. The distance from any point \((x,y)\) on the parabola to the focus \((1,-7)\) and to the directrix \(x = -5\) is equal. The distance to the focus is \(\sqrt{(x-1)^2 + (y+7)^2}\) and the distance to the directrix is \(|x+5|\).

Step 3: Set the Distances Equal

Set the distance to the focus equal to the distance to the directrix: \[ \sqrt{(x-1)^2 + (y+7)^2} = |x+5| \]

Step 4: Simplify the Equation

Square both sides to eliminate the square root:\[ (x-1)^2 + (y+7)^2 = (x+5)^2 \]Expand these expressions to get:\[ x^2 - 2x + 1 + y^2 + 14y + 49 = x^2 + 10x + 25 \]

Step 5: Rearrange and Solve

Simplify and rearrange the equation:\[ -2x + 14y + 50 = 10x + 25 \]Combine like terms:\[ 14y = 12x - 25 \]Finally, solve for \(y\) to express it in terms of \(x\):\[ y = \frac{6}{7}x - \frac{25}{14} \]

Step 6: Confirm the Parabola Orientation

Since the directrix is a vertical line \(x = -5\) and the focus is at \((1,-7)\), we have a horizontally oriented parabola. The equation confirms this orientation.

Key Concepts

Focus and Directrix

In the study of parabolas, two important terms often come up: the focus and the directrix. These are pivotal in comprehending the geometrical and algebraic setup of a parabola.
The **focus** is a point which, together with the directrix, defines the parabola. For a given point on the parabola, the focus is equally distanced as it is from the directrix. In our problem, the focus is located at the coordinates \(1, -7\), serving as a fixed point from which we measure distance.
The **directrix** is a line that is perpendicular to the line through the focus and parallel to the axis of symmetry of the parabola. In this case, it is given by the vertical line equation \(x = -5\). The directrix doesn't run through the parabola but stands as a boundary against which distances are measured.
Understanding these components is crucial:

• The focus is always above, below, or to the side (depending on orientation) of the parabola.
• The directrix never touches the parabola.
• The property of equidistance to the focus and directrix defines the parabola's shape.

Both of these elements are foundational in defining the equation of a parabola and help determine its orientation and position.

Horizontally Oriented Parabola

A parabola's orientation can be horizontal or vertical, and this orientation plays a significant role in its equation form.
An interesting detail about horizontally oriented parabolas is that they open sideways (left or right), unlike the more commonly seen vertical ones which open up or down. In this exercise, a horizontally oriented parabola is indicated by the vertical directrix line, \(x = -5\), and the focus at \(1, -7\).
Here are a few points about horizontally oriented parabolas:

• The directrix is vertical, while for vertical parabolas, it would be horizontal.
• The focal distance helps decide the width and position **on** the axis.
• Given the location of the directrix and focus, the parabola in this problem faces right.

Understanding the parabola’s orientation helps simplify solving its equation. This form of a parabola is defined by its equation, typically written as \(y = ax^2 + bx + c\) when horizontal transformations are applied.

Parabola Definition

At its core, a parabola is defined as a set of points in a plane equidistant from a given point, known as the focus, and a given line, known as the directrix.
In mathematical terms, the parabola is described by the equation resulting from this equidistance property. This equation can manifest itself either horizontally or vertically, depending on the line and point's relative positions, as illustrated in our exercise.
Here's an easy way to grasp the concept:

• The equidistance \(d\) property is described with a focus \(F(h, k)\) and directrix \(l: x = a\) for horizontal parabolas.
• Use the equation \[\sqrt{(x-h)^2 + (y-k)^2} = |x-a|\]
• Simplify this to solve for \(y\) or \(x\) to obtain the parabola's equation.

The elegance of a parabola's definition lies in its simple yet profound geometric property: the balance between its components leads to its characteristic "U" or "C" shape depending on the orientation. Appreciating this often overlooked aspect helps students not only solve problems but also visualize them in a broader mathematical context.