Problem 10 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

Focus: \(\left( \frac{3}{4}, 0 \right)\), Directrix: \(x = -\frac{3}{4}\), Focal Diameter: 3.

Step 1: Identify the Parabola Form

The given equation is \( y^2 = 3x \). This equation suggests that the parabola opens to the right, as it is in the form \( y^2 = 4px \). Here, \( 4p = 3 \).

Step 2: Determine the Value of p

We compare \( 4p = 3 \) with our standard form \( 4px \). Solving for \( p \), we get \( p = \frac{3}{4} \).

Step 3: Find the Focus Coordinates

Since \( p = \frac{3}{4} \) and the parabola opens to the right, the focus point \((p, 0)\) is \( \left( \frac{3}{4}, 0 \right) \).

Step 4: Identify the Directrix Equation

The directrix of a parabola that opens to the right is a vertical line at \( x = -p \). Therefore, the directrix is \( x = -\frac{3}{4} \).

Step 5: Calculate the Focal Diameter

The focal diameter of a parabola is the absolute value of \( 4p \). Since \( 4p = 3 \), the focal diameter is \( 3 \).

Step 6: Sketch the Parabola

The sketch of the parabola is as follows: Draw a U-shaped curve opening to the right, with the vertex at the origin \((0,0)\). Place the focus at \( \left( \frac{3}{4}, 0 \right) \) and draw the directrix as the vertical line at \( x = -\frac{3}{4} \). The parabola is symmetric about the x-axis.

Key Concepts

Focus of a Parabola

The focus of a parabola is a crucial point that helps define its shape and direction. In a parabola that opens to the right, like the equation \(y^2 = 3x\), the focus lies along the x-axis at a distance \(p\) from the vertex (the origin in this case). Given that \(4p = 3\), we find \(p\) by solving \(p = \frac{3}{4}\). The coordinates of the focus for this parabola are \(\left( \frac{3}{4}, 0 \right)\). This means the focus is 0.75 units to the right of the origin. The presence of the focus in a parabola ensures that all the points on the curve are equidistant from the focus and the directrix (which we'll discuss in the next section). This unique property makes the parabola highly reflective, often used in satellite dishes and headlight reflectors.

Directrix of a Parabola

The directrix of a parabola is an important geometric concept that complements the focus. It is a line, and for a right-opening parabola like \(y^2 = 3x\), the directrix is a vertical line. The equation of the directrix can be obtained by taking the negative value of \(p\), thus it lies at \(x = -\frac{3}{4}\).This line is parallel to the y-axis, exactly \(\frac{3}{4}\) units to the left of the origin. Points on the parabola are equidistant from both this line and the focus. Understanding the placement of the directrix helps in accurately sketching the graph of the parabola, ensuring that it is correctly oriented and symmetrically formed.

Focal Diameter

The focal diameter, sometimes called the "latus rectum," is the width of the parabola at its widest point, which is at the level of the focus. For the given equation \(y^2 = 3x\), we determine the focal diameter by evaluating \(|4p|\). Here, \(4p = 3\), so the focal diameter is \(3\).What this means in simple terms is that the parabola spans 3 units directly across its axis of symmetry, centered at the focus. This measurement is crucial for fully understanding the scale and scope of the parabola's curvature. Knowing the focal diameter also aids in applications which require precision, such as lens crafting and telescope design.