Question

For the following exercises, find the Cartesian equation describing the given shapes. A parabola with focus \((2,-5)\) and directrix \(x=6\)

Short answer

The Cartesian equation is \((x-4)^2 = 8(y+5)\).

Step-by-step solution

Understand the Definition

A parabola is defined as the set of all points equidistant from a point called the focus and a line called the directrix.

Identify the Given Parameters

The given focus of the parabola is \((2, -5)\) and the directrix is the line \(x = 6\). This indicates that the parabola opens horizontally since the directrix is a vertical line.

Understand the Standard Form

For a horizontal parabola, the equation can be given by \[(x-h)^2 = 4p(y-k)\]where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus.

Find the Vertex and Parameter p

The vertex \((h, k)\) is midway between the focus and the directrix. The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-value of the directrix: \[h = \frac{2+6}{2} = 4\]Since the parabola opens horizontally, the y-coordinate remains the same as that of the focus: \(k = -5\).

Determine the Distance p

The distance \(p\) is half the distance between the focus and the directrix. Thus: \[p = \frac{|6 - 2|}{2} = 2\]

Write the Equation of the Parabola

Now substituting the values of \(h\), \(k\), and \(p\) into the standard form, the equation becomes: \[(x-4)^2 = 8(y+5)\]

Key concepts

Focus and Directrix

A parabola is a fascinating geometric shape that emerges in various scientific and mathematical contexts. At its core, a parabola is defined by its **focus** and **directrix**. The **focus** is a fixed point, and the **directrix** is a fixed line. What makes a parabola special is that every point on the parabola is equidistant from the focus and the directrix.

In our exercise, the given focus is \(2,-5\), and the directrix is \(x = 6\). Since the directrix is vertical, the parabola will open horizontally. This orientation influences the form of the equation that we'll use. Recognizing the placement and roles of these components aids in deriving the equation of the parabola.

• Focus: A point where distances are measured.
• Directrix: A line serving as a reference for distances.
• Parabola: The set of equidistant points from both the focus and the directrix.

Vertex of a Parabola

To locate the vertex of a parabola, you need to consider both the focus and the directrix. The **vertex** is the midpoint horizontally between the focus and the directrix. It's a pivotal point where the parabola shifts direction.

For our problem, since the directrix \(x = 6\) is vertical, the x-coordinate of the vertex is the arithmetic mean of the x-coordinate of the focus and the directrix's x-value: \[h = \frac{2 + 6}{2} = 4\]. The y-coordinate is the same as the focus to maintain this horizontal opening, making \(k = -5\). Therefore, the vertex of the parabola is located at \(4, -5\).

• Vertex: Central point of the parabola, halfway between focus and directrix.
• Influences the orientation and location of the parabola.

Cartesian Equation

The Cartesian equation of a parabola is a formula that represents all its points in the Cartesian coordinate system. For a horizontally oriented parabola, such as in our exercise, the standard form of the equation is \((x-h)^2 = 4p(y-k)\).

In this form:

• \(h, k\) represents the vertex's coordinates.
• \(p\) is the distance from the vertex to the focus, which also influences the width and direction of the parabola.

After identifying \(h = 4\), \(k = -5\), and \(p = 2\) from our focus \((2, -5)\) and directrix \(x = 6\), we can substitute these into the equation.

Thus, the Cartesian equation for the parabola is \((x-4)^2 = 8(y+5)\). This equation reveals how the parabola stretches and orients in the Cartesian plane depending on its focus and directrix.