Problem 21 - Precalculus Mathematics for Calculus

Question

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$5 x+3 y^{2}=0$$

Step-by-Step Solution

Answer

Focus: \((-\frac{3}{20}, 0)\); Directrix: \(x = \frac{3}{20}\); Focal diameter: \(\frac{3}{5}\).

Step 1: Write the equation in standard form

The given equation is \(5x + 3y^2 = 0\). Let's rearrange it to match the standard form of a parabola. First, solve for \(x\): \(5x = -3y^2\). Then divide both sides by 5 to get \(x = -\frac{3}{5}y^2\). This is now in the form \(x = 4py^2\), which is a horizontal parabola.

Step 2: Identify the value of \(p\)

By comparing \(x = -\frac{3}{5}y^2\) with the standard form \(x = 4py^2\), we can see that \(4p = -\frac{3}{5}\). Solving for \(p\), we get \(p = -\frac{3}{20}\).

Step 3: Find the focus of the parabola

For a horizontal parabola in the form \(x = 4py^2\), the focus is located at the point \((h + p, k)\), where \(h = 0\) and \(k = 0\) in our transformed equation. The focus is therefore \((-\frac{3}{20}, 0)\).

Step 4: Determine the directrix of the parabola

The directrix of a parabola \(x = 4py^2\) is given by \(x = h - p\). With \(h = 0\) and \(p = -\frac{3}{20}\), the equation of the directrix is \(x = \frac{3}{20}\).

Step 5: Calculate the focal diameter

The focal diameter of a parabola is \(|4p|\). Substituting \(p = -\frac{3}{20}\), we get \(4p = -\frac{3}{5}\). The absolute value is \(\left|-\frac{3}{5}\right| = \frac{3}{5}\). Thus, the focal diameter is \(\frac{3}{5}\).

Step 6: Sketch the graph

Draw a horizontal parabola opening to the left, with its vertex at the origin \((0,0)\). Mark the focus at \((-\frac{3}{20}, 0)\) and draw the directrix line at \(x = \frac{3}{20}\). This visual represents the properties we calculated.

Key Concepts

Focus of a Parabola

The focus of a parabola is a unique point that helps define the parabola's shape and direction. Every point on the parabola is equidistant from the focus and the directrix, a concept called the "Reflective Property of Parabolas." This property is what allows parabolas to focus light or sound, making them useful in applications like satellite dishes and headlights.
To find the focus in our example, we noted that the parabola is open horizontally, and our equation is in the form of \(x = 4py^2\). In this standard form, the focus's coordinates are given by \((h+p, k)\), where \(h\) and \(k\) are the horizontal and vertical shifts, respectively. Here, since the vertex is at the origin, \(h = 0\) and \(k = 0\). We calculated \(p\) as \(-\frac{3}{20}\), hence the focus is at \((-\frac{3}{20}, 0)\).
Understanding the focus is crucial as it helps in sketching the parabola and knowing its symmetry axis.

Directrix of a Parabola

The directrix of a parabola is another key feature that works in tandem with the focus to define the properties of the parabola. It is a straight line that doesn't touch the parabola but serves as a reference point for its construction. In our case, the parabola is horizontal, meaning the directrix is a vertical line.
The standard formula states the directrix for \(x = 4py^2\) is \(x = h - p\). With \(h = 0\) and the previously calculated \(p = -\frac{3}{20}\), we find the directrix at \(x = \frac{3}{20}\).

• The directrix gives insight into how the parabola opens and helps in plotting its structure.
• It ensures that there's maintained distance symmetry with every point on the curve.

This concept helps provide a complete understanding of the geometric nature of a parabola.

Focal Diameter

Focal diameter, often called the latus rectum, is a segment that runs parallel to the directrix and passes through the focus of the parabola. This diameter is perpendicular to the axis of symmetry and reveals a lot about the spread of the parabola.
In mathematical terms, the focal diameter is given by \(|4p|\). Within our parabola, we know \(p\) is \(-\frac{3}{20}\), so \(4p\) equals \(-\frac{3}{5}\). The focal diameter, therefore, becomes \(\left| \frac{-3}{5} \right| = \frac{3}{5}\).

• The focal diameter not only indicates how wide the parabola opens but also helps in perfecting the plot by determining how far the sides or arms of the parabola extend.
• It is a measure related to the parabola's "thickness" relative to how the parabola is oriented around its vertex.

Understanding this measure makes it easier to precisely sketch the parabola.

Standard Form of Parabola Equation

The standard form of a parabola equation acts as a tool to transform and simplify equations for better understanding and manipulation. For our example, we started with the equation \(5x + 3y^2 = 0\) and converted it to its standard form \(x = 4py^2\).

• This form helps identify the key elements of a parabola, such as the focus, directrix, and orientation (horizontal or vertical).
• In our converted equation, \(p = -\frac{3}{20}\) informed us about the parabola's direction, because whether \(p\) is positive or negative hints at whether the parabola opens to the right or left, respectively.
• Using this form allows us to reliably extract and use specific parameters to explore the graphs' behavior fundamentally.

Transforming equations into their standard form is essential to mastering the details and geometrical features of parabolas.