Problem 59 - Precalculus with Limits

Question

In Exercises 51-60, find the standard form of the equation of the parabola with the given characteristics. Focus: \((2, 2) \quad\) directrix: \(x=-2\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with focus (2, 2) and directrix x=-2 is \( (y-2)^2=16(x-2) \).

Step 1: Determine the Direction of Opening

To know if the parabola opens left or right, look at where the focus and directrix are. In this case, the focus is to the right of the directrix which means that the parabola opens to the right. The standard form for a parabola that opens to the right is \( (y-k)^2=4p(x-h) \).

Step 2: Determine Vertex (h, k)

The vertex of the parabola is the point halfway between the focus and the directrix. Since the directrix is a vertical line \( x=-2 \), and the x-coordinate for the focus is 2, the x-coordinate of the vertex (h) would be \( h=(2-(-2))/2=2 \). We don't have any information that gives us the y-coordinate of the vertex directly. But we know that it remains the same as the y-coordinate of the focus. Therefore, the y-coordinate of the vertex k is equal to the y-coordinate of the focus, which is 2. Therefore, the vertex (h, k) equals (2, 2).

Step 3: Determine the Value of p

The value of p is the distance from the vertex to the focus or the directrix. Since the coordinate of the directrix is \( x=-2 \) and the x-coordinate of the vertex is 2, the distance p is \( p=2-(-2)=4 \). Therefore, p=4.

Step 4: Substitute (h, k) and p into the formula

Substitute h, k, and p into the standard form equation. We have \( h=2, k=2, \) and \( p=4 \). Substituting these values into the equation gives us \( (y-2)^2=4*4(x-2) \). Simplifying the equation gives \( (y-2)^2=16(x-2) \).

Key Concepts

Vertex

The vertex is the turning point of a parabola. It is crucial because it gives us the exact center of the parabola where it changes direction. In our problem, the vertex is found between the focus and the directrix.

Here's how you calculate it:

• Locate the x-coordinates of both focus and directrix: For our exercise, the focus is at (2, 2) and the directrix is at the vertical line \(x = -2\).
• Compute the midpoint between these x-coordinates: \(h = \frac{2 + (-2)}{2} = 0\).
• The y-coordinate of the vertex remains the same as the y-coordinate of the focus which is 2.
• Thus, the vertex \((h, k)\) is \((0, 2)\).

This method ensures you accurately find the vertex, essential for plotting the parabola.

Focus

The focus is a point inside the parabola that is key to its shape and direction. The parabola curves in a way that every point on it is equidistant to the focus and the directrix.

For our specific parabola:

• The focus is at \((2, 2)\).
• It lies horizontally from the directrix, which is why our parabola opens to the right.
• The distance between the focus and the vertex is the absolute value of \(p\), key for the parabola's equation.

Understanding the focus helps grasp why the parabola is shaped as it is and provides insight into its real-world applications, such as satellite dishes and headlights.

Directrix

The directrix is a fixed line used to establish the exact shape of the parabola. Serving as a reference point, it aids in defining the parabola's orientation and curvature.

Here are some essential details:

• In our problem, the directrix is \(x=-2\).
• As the parabola opens to the right, the directrix tells us that the vertex must lie 4 units to the left of the focus.
• This distance is crucial as it equals the value of \(p\), the coefficient that defines the parabola's equation \((y-k)^2=4p(x-h)\).

The directrix is the guiding line that, along with the focus, uniquely determines the parabola's path.