Problem 16 - Precalculus: Mathematical for Calculus

Question

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x-7 y^{2}=0$$

Step-by-Step Solution

Answer

Focus: \(\left(\frac{7}{4}, 0\right)\), Directrix: \(x = -\frac{7}{4}\), Focal diameter: 7.

Step 1: Bring the equation to standard form

The given equation is \(x - 7y^2 = 0\). To bring this equation into the standard form of a parabola, we rearrange it as \(x = 7y^2\).

Step 2: Identify the standard form

The standard form of a parabola that opens horizontally is \((x - h) = 4p(y - k)^2\). Comparing this with \(x = 7y^2\), we can see it's of the form \(x = 4p(y - 0)^2\) where \(4p = 7\), thus \(h = 0\) and \(k = 0\).

Step 3: Calculate the value of \(p\)

From \(4p = 7\), we solve for \(p\) by dividing both sides by 4. Thus, \(p = \frac{7}{4}\).

Step 4: Find the focus

The focus of a parabola \((x - h) = 4p(y - k)^2\) is at \((h + p, k)\). Since \(h = 0\), \(k = 0\), and \(p = \frac{7}{4}\), the focus is at \(\left(\frac{7}{4}, 0\right)\).

Step 5: Determine the directrix

The directrix of this horizontal parabola is \(x = h - p\). Substituting \(h = 0\) and \(p = \frac{7}{4}\), the directrix is \(x = -\frac{7}{4}\).

Step 6: Calculate the focal diameter

The focal diameter is the absolute value of \(4p\). Since \(4p = 7\), the focal diameter is 7.

Step 7: Sketch the Graph

Since the parabola is in the form \(x = 7y^2\), it opens to the right. The vertex at \((0,0)\), a focus at \(\left(\frac{7}{4}, 0\right)\), and a directrix of \(x = -\frac{7}{4}\) guide the sketch. The curve will be symmetrical about the x-axis.

Key Concepts

Focus

The focus of a parabola is a critical point that helps define its shape and orientation. For parabolas, the focus lies inside the curve and serves as one of the special geometric properties that every point on the parabola has. Understanding the focus is essential for graphing the parabola and for grasping its properties.

In the equation provided, \( x = 7y^2 \), the parabola opens horizontally. The focus is found using the formula

• Focus: \((h + p, k)\)

where \( h \) and \( k \) are the vertex coordinates, and \( p \) is the distance from the vertex to the focus along the axis of symmetry. In our case, \( h = 0 \) and \( k = 0 \).

The calculation for \( p \) from the equation \( 4p = 7 \) yields \( p = \frac{7}{4} \).
The focus is therefore located at \( \left(\frac{7}{4}, 0\right) \). This means the focus is just a little to the right of the vertex, which is located at the origin.

Directrix

A directrix is a straight line used to define a parabola along with its focus. In simple terms, the directrix helps in understanding how the parabola's shape is structured. Every point on the parabola is equidistant from the focus and this directrix.

In our horizontal parabola equation \( x = 7y^2 \), the directrix can be found using:

• Directrix: \( x = h - p \)

Given \( h = 0 \) and \( p = \frac{7}{4} \), the directrix is calculated as \( x = -\frac{7}{4} \).

This tells us that the directrix is a vertical line located at \( x = -\frac{7}{4} \). It's on the opposite side of the vertex from the focus. This line doesn't physically touch the parabola but is vital in its definition. Understanding that the directrix is at \( x = -\frac{7}{4} \) helps when sketching or analyzing the parabola.

Focal Diameter

The focal diameter, also known as the latus rectum, is a measure of the width of a parabola at the level of the focus. It's important for understanding both the size and proportion of the curve.

For the equation \( x = 7y^2 \), the focal diameter is given by the value \( 4p \), which can be directly calculated from the equation standard form. Since we know \( 4p = 7 \), the focal diameter is simply 7.

The focal diameter provides a sense of scale to the parabola's shape. In this example, the focal diameter being 7 indicates how spread out the parabola is at the focus. This value also represents the length of the line segment that is parallel to the directrix and passes through the focus.