Problem 258 - Calculus Volume 3

Question

Determine the equation of the parabola using the information given. Focus (2,3) and directrix \(x=-2\)

Step-by-Step Solution

Answer

The equation is \((y-3)^2 = 8x\).

Step 1: Understand the Parabola Components

A parabola is defined as the set of all points equidistant from a point called the "focus" and a line called the "directrix". In this exercise, the focus is at \((2,3)\) and the directrix is the line \(x = -2\).

Step 2: Recall the Parabola Equation for a Horizontal Axis

When a parabola opens horizontally (to the left or right), its equation is of the form \((y-k)^2 = 4p(x-h)\), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus (or directrix).

Step 3: Calculate Vertex and Distance p

The vertex is halfway between the focus and the directrix on the x-axis. The focus is at \(x = 2\) and the directrix at \(x = -2\). The vertex is at \(x = \frac{2 + (-2)}{2} = 0\) and \(y = 3\), so the vertex is \((0,3)\). The distance \(p\) is the distance from the vertex to the focus: \(p=2\).

Step 4: Construct the Parabola Equation

With \((h,k) = (0,3)\) and \(p = 2\), substitute these into the equation for a parabola opening to the right: \((y-3)^2 = 4 \times 2 \times (x-0)\). Simplify to get \((y-3)^2 = 8x\).

Step 5: Final Equation

The equation of the parabola with focus \((2,3)\) and directrix \(x=-2\) is \((y-3)^2 = 8x\).

Key Concepts

Parabola Components

Understanding the components of a parabola is essential in graphing it correctly. The main components of a parabola are:

• Focus: A point from which distances are measured in defining a parabola.
• Directrix: A line from which distances are measured in defining a parabola, opposite to the focus.
• Vertex: The point on the parabola that is equidistant from both the focus and the directrix. It acts as the turning point.

In the given exercise, the focus is (2,3) and the directrix is the line (x = -2) . Recognizing these components helps in situating the parabola spatially.

Focus and Directrix

The focus and directrix are critical in defining a parabola's shape and direction.
The focus, located at a specific point, influences how the parabola opens and bends.
Meanwhile, the directrix is pivotal in determining the parabola's location relative to the axis. In this exercise:

• The focus is situated at (2,3) , meaning it lies in the coordinate plane's positive x-axis section.
• The directrix is the vertical line (x = -2) , serving as a boundary that essentially helps "pull" the parabola open.

These two together enable the calculation of the vertex and the parabola's precise path.

Vertex Calculation

The vertex is arguably the most critical point in a parabola, marking the spot where it changes direction.
To find the vertex, determine the midpoint between the focus and the directrix along the x-axis.Here, the focus is at (2,3) and the directrix is (x = -2). Calculating the x-coordinate of the vertex involves:\[x = \frac{2 + (-2)}{2} = 0\]The y-coordinate remains the same as the focus, so the vertex lies at (0,3).
This vertex becomes the reference point for placing the rest of the parabola.

Equation of Parabola

Understanding the equation of a parabola is crucial to representing it algebraically. For horizontal parabolas, the equation is derived from:\[(y-k)^2 = 4p(x-h)\]Here, (h,k) is the vertex, and (p) represents the distance from the vertex to the focus or the directrix. In this exercise, the vertex is (0,3) and (p) is 2, as the focus and vertex are separated by 2 units.To form the equation, substitute these into the standard form:

• Vertex: (h,k) = (0,3)
• Distance:(p) = 2

Plug in the values:\[(y-3)^2 = 4 \times 2 \times (x-0)\]Simplifying gives you the final equation:\[(y-3)^2 = 8x\]This equation locates the horizontal parabola turning point and its opening to the right, guiding how it plots on a graph.