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Chapter 10: Problem 33

Write an equation for each parabola with vertex at the origin. focus \((5,0)\)

Short Answer

Expert verified
The equation is \[ x = \frac{1}{20} y^2 \]

Step by step solution

01

Understanding the Parabola's Properties

The parabola's vertex is at the origin, \(0,0\). The focus is given at \(5,0\). This indicates that the parabola opens to the right since the focus lies on the positive x-axis relative to the vertex.
02

Using the Vertex and Focus Relationship

The standard form for a parabola that opens to the right with vertex at the origin is \(x = \frac{1}{4p}y^2\), where \(p\) is the distance from the vertex to the focus. Here, \(p = 5\).
03

Substituting the Value of p

Substitute \(p = 5\) into the equation: \[x = \frac{1}{4 \times 5} y^2 = \frac{1}{20} y^2 \]
04

Final Equation

The equation of the parabola with vertex at the origin and focus at \(5,0\) is \[ x = \frac{1}{20} y^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex at the origin
A parabola's vertex is the point where it changes direction. In this exercise, the vertex is at the origin, which means the coordinates are \(0,0\). This makes the calculations straightforward. Understanding vertices is key when dealing with parabolas as they serve as a reference point for defining the shape. For parabolas, the vertex is also the midpoint between the focus and the directrix.
focus of parabola
The focus of a parabola is a crucial point. It defines the reflective property unique to parabolas. If the vertex is at the origin, the focus \(5,0\) tells us that the parabola opens rightwards along the x-axis. The distance \(p\) from the vertex to the focus is fundamental. Here, \(p = 5\). This distance affects the 'width' of the parabola. For parabolas that open left or right, the equation in standard form involves \(p\) like so: \[ x = \frac{1}{4p} y^2 \]. This helps us understand the shape and orientation of the parabola from just the focus.
standard form of parabola
The standard form of the equation for a parabola depends on its orientation. When the vertex is at the origin and the parabola opens sideways, the equation generally looks like \[ x = \frac{1}{4p}y^2 \]. In this scenario, since \(p = 5\), we substitute \(p\) into the standard form: \[ x = \frac{1}{4 \times 5} y^2 = \frac{1}{20} y^2 \]. This equation tells us that for every value of \(y\), \(x\) changes by a fraction due to the \( \frac{1}{20} \) factor, which describes how wide or narrow the parabola is. Understanding this standard form is essential for graphing and analyzing the properties of parabolas.

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Most popular questions from this chapter

Find the eccentricity of each hyperbola to the nearest tenth. $$\frac{x^{2}}{2}-\frac{y^{2}}{18}=1$$

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