Problem 261 - Calculus Volume 3

Question

Determine the equation of the parabola using the information given. Focus (-3,5) and directrix \(y=1\)

Step-by-Step Solution

Answer

The equation of the parabola is \((x + 3)^2 = 8(y - 3)\).

Step 1: Interpret the Given Information

We have a focus at the point (-3, 5) and a directrix at the line y=1. This means that our parabola is vertical because the directrix is a horizontal line (y = constant value).

Step 2: Calculate the Vertex

The vertex lies midway between the focus and the directrix. The y-coordinate of the vertex can be calculated by averaging the y-coordinates of the focus and directrix. Thus, the y-coordinate of the vertex will be: \( \frac{5+1}{2} = 3 \). The x-coordinate remains the same as that of the focus, which is -3. Hence, the vertex is at (-3, 3).

Step 3: Determine the Distance to Focus (\(p\))

The distance \( p \) is the distance from the vertex to the focus. Since the vertex is at (-3, 3) and the focus is at (-3, 5), \( p = 5 - 3 = 2 \).

Step 4: Write the Parabola Equation

For a vertical parabola with vertex at \((h, k)\) and distance \(p\), the equation is \((x - h)^2 = 4p(y - k)\). Here, \( h = -3 \), \( k = 3 \), and \( p = 2 \). Substitute these values into the equation: \((x + 3)^2 = 8(y - 3)\).

Key Concepts

parabola equation

To understand how to derive the equation of a parabola, it's important to grasp the standard form of a parabolic equation. For a vertical parabola, the standard form is given by \[(x - h)^2 = 4p(y - k)\] In this equation,

• \((h, k)\) represents the vertex of the parabola.
• \(p\) is the distance from the vertex to the focus.

This particular formulation highlights the symmetry of the parabola, indicating that any point (x, y) on the parabola is equidistant from the focus and the directrix. For our exercise, with the vertex at (-3, 3) and \(p\) being equal to 2 (the distance from the vertex to the focus), the equation becomes \[(x + 3)^2 = 8(y - 3)\]. This encompasses the structural nature of the parabola defined by its geometric properties.

focus and directrix

A parabola is uniquely defined by its focus and directrix. Here's how they work:

• The focus is a fixed point inside the parabola. It serves as a point of reference for the shape of the parabola.
• The directrix is a line perpendicular to the axis of symmetry of the parabola.

The fundamental property of a parabola is this: any point on the parabola is equidistant from the focus and the directrix. In our exercise, the focus is at (-3, 5), and the directrix is the horizontal line y = 1. This configuration confirms that the parabola opens vertically, as the directrix is horizontal. This geometric property not only defines the direction the parabola opens but also helps determine other key features like the vertex and the equation itself.

vertex of a parabola

The vertex of a parabola is a critical point as it gives us a sense of orientation and location on the coordinate plane. It is found at the exact midpoint between the focus and the directrix. To locate the vertex, perform a simple averaging of the y-coordinates of the focus and directrix for vertical parabolas. This yields: \[y_{vertex} = \frac{y_{focus} + y_{directrix}}{2} = \frac{5 + 1}{2} = 3\] The x-coordinate remains unchanged from the focus, keeping it -3. So, the vertex is at (-3, 3). This point not only anchors the parabola's graph but is also pivotal to writing the parabola's equation. By knowing the vertex, we can properly align the parabola on the geometric plane, ensuring it symmetrically straddles its axis (a vertical or horizontal line, depending on parabola orientation).