Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$ y^{2}=x $$

Step-by-Step Solution

Answer

Focus: \((\frac{1}{4}, 0)\); Directrix: \(x = -\frac{1}{4}\).

Step 1: Identify the Parabola Form

To solve for the coordinates of the focus and the equation of the directrix, begin by identifying the form of the given parabola equation. The parabola given is \( y^2 = x \), which is a horizontal parabola. It can be compared to the standard form of horizontal parabolas: \((y-k)^2 = 4p(x-h)\). In our equation, \(h = 0\), \(k = 0\), and \(4p = 1\).

Step 2: Solve for p

Recall that in the standard form \((y-k)^2 = 4p(x-h)\), \(4p\) represents the coefficient of \(x\). Here, \(4p = 1\), so solve for \(p\) by dividing: \(p = \frac{1}{4}\).

Step 3: Find the Focus

The formula for the focus of a horizontal parabola \((y-k)^2 = 4p(x-h)\) is \((h+p, k)\). For this equation, \(h = 0\), \(k = 0\), so the focus is \((\frac{1}{4}, 0)\).

Step 4: Find the Directrix

The directrix of a horizontal parabola \((y-k)^2 = 4p(x-h)\) is given by the equation \(x = h-p\). In this case, \(h = 0\), so the directrix is \(x = -\frac{1}{4}\).

Step 5: Sketch the Parabola and Label

Draw a sketch of the parabola with its vertex at the origin, opening to the right. Label the focus at \((\frac{1}{4}, 0)\) and the directrix as the vertical line \(x = -\frac{1}{4}\). This will help in visualizing the relationship between the parabola, its focus, and directrix.

Key Concepts

Parabola Equations

Parabolas are an essential part of coordinate geometry, often described as the locus of points that are equidistant from a fixed point, the focus, and a line, the directrix. The equation of a parabola in its standard form helps us identify its key features. When working with parabolas, it's crucial to recognize the role of the vertex form:

• Vertical parabolas: \(x-h\)^2 = 4p\(y-k\)
• Horizontal parabolas: \(y-k\)^2 = 4p\(x-h\)

These equations allow us to find the parabola's orientation and its important elements like the focus and directrix.
In the equation \(y^2 = x\), we can infer that it is a horizontal parabola since the squared term is \(y^2\). By comparing it to the standard form, \(h\) and \(k\) are both zero, and \(4p = 1\) gives us the value of \(p\), which we use to find the focus and directrix.

Conic Sections

Parabolas form part of a fascinating group known as conic sections. These are curves obtained by intersecting a cone with a plane. Besides parabolas, conic sections include circles, ellipses, and hyperbolas. Each conic has distinct properties and equations.

• Circle: \(x^2 + y^2 = r^2\)
• Ellipses: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
• Hyperbolas: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Parabolas: Described by \(ax^2 + by^2 + cx + dy + e = 0\)

Each type of conic section can be identified by examining its general equation and the values of the coefficients. Parabolas specifically can be recognized by having only one squared term, making them unique among their conic relatives.

Coordinate Geometry

Understanding parabolas through the lens of coordinate geometry allows us to analyze their geometric properties in the context of a coordinate plane. By using the standard equation, we can derive essential details about the parabola such as its vertex, focus, and directrix.
Parabolas have a plethora of applications:

• Designing paths of projectiles in physics
• Optimizing satellite dishes for maximum signal reception
• Determining areas of plots in architecture

Coordinate geometry provides tools to graph and visualize these curves as well as solve real-world problems. It enables us to describe precisely where a parabola sits in relation to other geometric structures or points in the plane.

Horizontal Parabola

When assessing whether a parabola is horizontal or vertical, we examine the placement of the squared term in its equation. For horizontal parabolas, the \(y\) variable is squared, which dictates how the parabola opens in the coordinate plane.
In the case of the equation \(y^2 = x\), this indicates a horizontal reflection, opening to the right, with the vertex at the origin. From the equation, you also derive:

• Focus: \(\left( \frac{1}{4}, 0 \right)\) - located to the right of the vertex
• Directrix: \(x = -\frac{1}{4}\) - a vertical line left of the vertex

These components are vital in defining the parabola's geometry and ensuring accurate graphs and understanding of its orientation in space.