Problem 16 - Algebra and Trigonometry

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: Identify the Parabola's Equation Form

The given equation is \( x - 7y^2 = 0 \). To find the focus and directrix, we need it in the standard form of a parabola. This equation can be rewritten as \( x = 7y^2 \), which is similar to the standard form \( x = 4py^2 \).

Step 2: Determine Parameter 'p'

To rewrite \( x = 7y^2 \) in the form \( x = 4py^2 \), compare the coefficients. Here, \( 4p = 7 \). Solving for \( p \) gives \( p = \frac{7}{4} \).

Step 3: Find the Focus

For a parabola in the form \( x = 4py^2 \), the focus is positioned horizontally from the vertex, at \((p, 0)\). Using \( p = \frac{7}{4} \), the focus is \( \left( \frac{7}{4}, 0 \right) \).

Step 4: Determine the Directrix

The directrix of the parabola is the vertical line \( x = -p \), which is \( x = -\frac{7}{4} \).

Step 5: Calculate the Focal Diameter

The focal diameter (or latus rectum) is given by \( |4p| \). Substituting \( p = \frac{7}{4} \), the focal diameter is \( 7 \).

Step 6: Sketch the Graph

To sketch the graph, plot the vertex at the origin \((0, 0)\), note the focus at \( \left( \frac{7}{4}, 0 \right) \) to the right of the vertex, and draw the directrix as the line \( x = -\frac{7}{4} \). The parabola opens to the right because the coefficient of \( y^2 \) is positive and it will be symmetric with respect to the x-axis.

Key Concepts

Focus of a Parabola

In a parabola, the focus is a special point that plays a significant role in its geometry. Imagine the parabola as a curved mirror, and the focus is the spot where light or sound, if directed towards the parabola, will converge. In mathematical terms, the focus is a fixed point from which the distance to any point on the parabola is equal to the distance from that point to a line called the directrix. For the parabola given by the equation \(x = 7y^2\), it corresponds to the standard form \(x = 4py^2\).

Here, by comparing, we find \(4p = 7\), which tells us that \(p = \frac{7}{4}\). This parameter \(p\) helps locate the focus. For this equation, the focus is located at \(\left( \frac{7}{4}, 0 \right)\).

The focus isn’t just a random point. It is vital because it defines how the parabola opens and its symmetry. Knowing the focus allows us to analyze and sketch the parabola much more accurately.

Directrix of a Parabola

The directrix of a parabola is a straight line that, in combination with the focus, defines the shape and orientation of the parabola. Every point on the parabola is equidistant from the focus and the directrix. This balance creates the unique shape of a parabola that we see.

For the parabola with the equation \(x = 7y^2\), the directrix is given by the vertical line \(x = -p\). Since \(p = \frac{7}{4}\), the equation of the directrix is \(x = -\frac{7}{4}\).

Understanding where the directrix is helps in visualizing the physical space the parabola occupies. It also assists in ensuring that our parabola sketch maintains its symmetry, as it will always be equidistant from the focus and directrix lines.

Focal Diameter

The focal diameter (also known as the latus rectum) is the length of a segment parallel to the directrix that passes through the focus and has its endpoints on the parabola. This is a critical factor that helps describe the width of the parabola. By measuring it, we can better grasp the size of the opening of the parabola.

For the equation \(x = 7y^2\), the focal diameter can be calculated using the formula \(|4p|\). Since we previously determined \(p = \frac{7}{4}\), the focal diameter is \(|4 \times \frac{7}{4}| = 7\).

This value tells us how wide the parabola is at its focus. Knowing the focal diameter is particularly useful because it allows us to figure out how much the parabola flares outward, giving a fuller understanding of its shape.