Problem 37
Question
Use the definitions of conic sections to answer the following. Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points \((5,0)\) and \((-5,0)\) is 14.
Step-by-Step Solution
Verified Answer
The conic section is an ellipse.
1Step 1: Recognize the Conic Section
A conic section defined by the condition that the sum of the distances from any point on it to two fixed points (the foci) is constant is an ellipse. In this problem, the fixed points are given as \((5,0)\) and \((-5,0)\), and the constant sum of distances is 14.
2Step 2: Calculate the Distance Between the Foci
The distance between the two foci, \((5,0)\) and \((-5,0)\), is calculated using the distance formula:\[ \text{Distance} = \sqrt{(5 - (-5))^2 + (0 - 0)^2} = \sqrt{(10)^2} = 10. \]This distance is less than the given constant sum, which confirms the shape is an ellipse.
3Step 3: Conclusion
Since the sum of the distances from any point on the conic section to the foci \((5,0)\) and \((-5,0)\) is constant and greater than the distance between the foci (which is 10), the conic section is an ellipse by definition.
Key Concepts
EllipseDistance FormulaFociEducational Problem Solving
Ellipse
An ellipse is a type of conic section, which is a curve formed by intersecting a plane with a cone. It is defined as the set of points in a plane where the sum of the distances to two fixed points, called foci, is constant. This distinguishes it from other conic sections, such as circles, hyperbolas, and parabolas.
Ellipses appear in many fields, including astronomy, where planets' orbits around the sun are elliptical. The major and minor axes are the principal axes of an ellipse, where the major axis is the longest diameter and the minor axis is the shortest. Knowing how to identify and work with ellipses is essential for understanding complex geometry concepts and applications.
Ellipses appear in many fields, including astronomy, where planets' orbits around the sun are elliptical. The major and minor axes are the principal axes of an ellipse, where the major axis is the longest diameter and the minor axis is the shortest. Knowing how to identify and work with ellipses is essential for understanding complex geometry concepts and applications.
Distance Formula
The distance formula is an important tool in geometry that helps find the distance between two points in a coordinate system. Given two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance between them is determined using the formula:
- \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula is derived from the Pythagorean theorem and is crucial for various calculations involving length and space, such as determining the size of line segments in both algebra and calculus contexts.
In our exercise, the distance formula helps verify the configuration of the ellipse by finding the distance between the foci and relating it to the constant sum of the distances to any point on the ellipse.
- \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula is derived from the Pythagorean theorem and is crucial for various calculations involving length and space, such as determining the size of line segments in both algebra and calculus contexts.
In our exercise, the distance formula helps verify the configuration of the ellipse by finding the distance between the foci and relating it to the constant sum of the distances to any point on the ellipse.
Foci
Foci are integral to understanding ellipses. Each ellipse has two such points, and their role is central in determining its shape. The sum of the distances from any point on the ellipse to the two foci remains constant, a defining property of ellipses.
In our problem, the foci are located at \( (5,0) \) and \( (-5,0) \).
The importance of foci can be seen in data analysis, architectural design, and physics, where they provide insight into system behaviors and relationships.
Understanding the position and function of the foci is key to solving problems involving ellipses, as demonstrated in this exercise.
In our problem, the foci are located at \( (5,0) \) and \( (-5,0) \).
The importance of foci can be seen in data analysis, architectural design, and physics, where they provide insight into system behaviors and relationships.
Understanding the position and function of the foci is key to solving problems involving ellipses, as demonstrated in this exercise.
Educational Problem Solving
Educational problem solving involves applying known principles and formulas, like those of conic sections, to determine unknown properties or confirm existing hypotheses. Leveraging steps such as recognizing patterns, applying mathematical formulas, and verifying results, equips students with a structured approach to analyzing and resolving problems.
The given exercise exemplifies this approach by:
The given exercise exemplifies this approach by:
- Recognizing and categorizing the conic section as an ellipse based on given conditions.
- Applying the distance formula to validate characteristics of the ellipse.
- Concluding through logical reasoning why certain properties define the geometric shape.
Other exercises in this chapter
Problem 36
Graph each circle by hand if possible. Give the domain and range. $$(x-4)^{2}+(y-3)^{2}=25$$
View solution Problem 37
Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) .$$x=t+2, y=\frac{1}{t+2}, \text { for } t \neq-2$$
View solution Problem 37
Find an equation for each ellipse. Vertices \((4,9)\) and \((4,1) ;\) minor axis of length 6
View solution Problem 37
Graph each circle by hand if possible. Give the domain and range. $$(x+3)^{2}+(y+2)^{2}=36$$
View solution