Problem 13 - Algebra and Trigonometry

Question

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=-8 y^{2}$$

Step-by-Step Solution

Answer

The focus is at \((-2, 0)\), the directrix is \(x = 2\), and the focal diameter is 8.

Step 1: Identify the Standard Form of the Parabola

The given equation is \(x = -8y^2\). This represents a parabola that opens horizontally. The standard form for a horizontally opening parabola is \(x = a(y - k)^2 + h\), where \( (h, k)\) is the vertex of the parabola, and \(a\) is a constant.

Step 2: Extract the Coefficient and Identify the Direction

From the equation \(x = -8y^2\), we deduce that \(a = -8\). Since \(a\) is negative, the parabola opens to the left. Here, the vertex \((h, k)\) is at the origin \((0, 0)\), as there are no added constants to \(x\) or \(y\).

Step 3: Calculate the Focus and Directrix

The relationship between \(a\) and the focus and directrix is given by \(4p = a\), where \(p\) is the distance from the vertex to the focus and directrix. For this equation, \(a = -8\), thus \[ 4p = -8 \implies p = -2. \]Since the parabola opens to the left, the focus lies on the left of the vertex, meaning the focus is at \((-2, 0)\) and the directrix is a vertical line on the right of the vertex, \(x = 2\).

Step 4: Determine the Focal Diameter

The focal diameter is the length of the segment through the focus that is perpendicular to the axis of symmetry. The focal diameter is given by the absolute value \(|a|\), thus:\[|a| = 8.\]

Step 5: Sketch the Parabola

To sketch the parabola:- Plot the vertex at \((0, 0)\).- Mark the focus at \((-2, 0)\) and draw the directrix as the line \(x = 2\).- Sketch the parabola opening leftward, ensuring that it passes through these coordinates and makes a symmetrical curve with respect to the focus and directrix.

Key Concepts

Parabola

The concept of a parabola is fundamental in understanding conic sections. A parabola is a symmetrical open curve, which can be thought of as the path traced by a point that moves such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix). This shape is prevalent in real-life applications, such as the path of a thrown object or the design of satellite dishes.

• Parabolas are either vertically or horizontally oriented, depending on the placement of their variable terms.
• The general property of a parabola is that it exhibits a U-shaped curve, and its sides are mirror images of each other.
• In terms of geometry, each point on a parabola is equidistant from the focus and the directrix.

Focus and Directrix

The focus and directrix are critical elements in defining the shape and orientation of a parabola.

• The focus is a point inside the parabola, and it lies along its axis of symmetry. This point is crucial as it helps determine the specific way a parabola opens or closes.
• The directrix, on the other hand, is a line that is perpendicular to the axis of symmetry. Contrary to the focus, it does not touch the parabola but acts as a reference line for accuracy and measurement.

For the equation of the parabola here, given as \(x = -8y^2\), the focus is found at \((-2, 0)\), indicating the parabola opens towards the left due to the negative coefficient. The directrix, in this scenario, is the line \(x = 2\), showcasing that both these components are aligned to aid in sketching and interpreting the parabola.

Standard Form of Parabola

The standard form of a parabola is crucial for identifying its vertex, axis of symmetry, and direction of opening. When dealing with conic sections, understanding how to write and manipulate these equations is necessary.

• For a horizontally oriented parabola, the standard form is: \(x = a(y - k)^2 + h\).
• If the parabola opens vertically, the form is \(y = a(x - h)^2 + k\).
• The vertex of the parabola is located at \((h, k)\), and this point is central to plotting the curve accurately.
• The constant \(a\) dictates the direction and width of the parabola. A positive \(a\) indicates an upward-facing parabola, while a negative value points down or to the left.

In our specific problem, \(x = -8y^2\) is aligned to open leftward, with the vertex situated at the origin \((0, 0)\). Understanding this helps visualize the parabola's structure and symmetry relative to its vertex.

Focal Diameter

The focal diameter is an important measure in understanding the extent of a parabola.

• Also known as the latus rectum, the focal diameter represents the length of a line segment that passes through the focus and is perpendicular to the axis of symmetry.
• In terms of calculation, it is equal to \(|a|\) from the parabola's equation. This indicates how 'wide' the parabola appears when viewed along its axis of symmetry.
• For the given equation \(x = -8y^2\), the focal diameter is \(8\), emphasizing the horizontal spread of the parabola.

Recognizing the focal diameter helps in accurately drawing the parabola and understanding its dimensions and proportions in the geometric plane.