Problem 8 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Focus: (0, 1/4); Directrix: y = -1/4; Focal Diameter: 1.

Step 1: Identify the Structure of the Parabola

The equation given is \(x^2 = y\), which can be rewritten as \(y = x^2\). This represents a parabola that opens upwards, where the equation is centered at the origin \((0,0)\).

Step 2: Determine the Parameters

In the equation of a parabola \(x^2 = 4py\), \(4p\) should equal 1 (matching \(x^2 = y\)). Therefore, \(p = \frac{1}{4}\). This parameter \(p\) provides details about the parabola's orientation and specific properties.

Step 3: Find the Focus

The focus of a parabola \(x^2 = 4py\) that opens upwards is located at \((0,p)\). Since \(p = \frac{1}{4}\), the focus is at \((0, \frac{1}{4})\).

Step 4: Locate the Directrix

The directrix of the parabola opening upwards is a horizontal line \(y = -p\). Therefore, the directrix is \(y = -\frac{1}{4}\).

Step 5: Calculate the Focal Diameter

The focal diameter is the length of the line segment perpendicular to the axis of symmetry that passes through the focus and whose endpoints lie on the parabola. It equals \(4p\), so in this case it is 1.

Step 6: Sketch the Graph

To sketch the graph of \(x^2 = y\), plot the focus at \((0, \frac{1}{4})\), draw the directrix as the line \(y = -\frac{1}{4}\), and sketch the parabola opening upwards passing through the origin.

Key Concepts

Focus and Directrix of a Parabola

A parabola is a U-shaped curve that has unique properties, including its focus and directrix.
The focus of a parabola is a point from which distances to any point on the parabola are measured equally with a corresponding line called the directrix. Together, they define the parabola.

• The focus acts like a magnet for the parabola's points, located at a specific distance within the curve.
• The directrix is a straight line that lies outside the parabola and is equidistant from all points on it to the focus.

The relationship between the focus and directrix helps to form the definition and shape of a parabola.
For our specific case, the parabola given by the equation \(x^2 = y\) has:

• Focus at point \((0, \frac{1}{4})\).
• Directrix as the line \(y = -\frac{1}{4}\).

This relationship between the focus and directrix shapes the upward-opening parabola through the origin.

Equation of a Parabola

The equation of a parabola is a mathematical expression that describes its position and curvature.
For vertical parabolas, the equation generally follows the form \(x^2 = 4py\).
Here, the parabola equation \(x^2 = y\) matches this format.

• The value \(4p\) in this equation usually expresses the distance to the focus and the length of the directrix line.

To understand this equation better, let's find \(p\) by setting \(4p = 1\) since \(x^2 = y\) becomes \(x^2 = 4py\). Solving gives \(p = \frac{1}{4}\).
The value of \(p\) directly influences the parabola's orientation and how far it opens. A larger \(p\) means a wider opening.
This specific equation represents a parabola:

• Opening upwards from its vertex at the origin \((0,0)\).
• With the knowledge of \(p\), we can deduce that focus is found at \((0, \frac{1}{4})\).
• The directrix is confirmed at \(y = -\frac{1}{4}\).

Focal Diameter

The focal diameter, also known as the latus rectum, is an important measure in a parabola.
It represents the width of the parabola at its focus and is directly linked to its parameter \(p\).
For any parabola with an equation \(x^2 = 4py\), the focal diameter equals \(4p\). Consider this as the breadth of the parabola through the focus intersecting the curve on both sides.

• This length provides insight into how "wide" the parabola is at the focus.

In our specific case with \(p = \frac{1}{4}\), we calculate this diameter:

• The focal diameter computes as \(4 \times \frac{1}{4} = 1\).

Thus, the parabola has a focal diameter of 1, indicating it is quite narrow at the location of the focus when compared to other values for \(p\).
This understanding of the focal diameter helps when sketching or analyzing the nature of the parabola.