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Sketch the two families of hyperbolas, the curves xy = k, for various real k and the curves x2 – y2 = m, for various real m. %3D Also show that these two families are orthogonal to each other at any point of intersection of a curve from one family with a curve from the other family. Here we avoid the origin, where there is a degeneracy. Also sketch the families on one graph.

Sketch the two families of hyperbolas, the curves xy = k, for various real k and the curves x2 – y2 = m, for various real m. %3D Also show that these two families are orthogonal to each other at any point of intersection of a curve from one family with a curve from the other family. Here we avoid the origin, where there is a degeneracy. Also sketch the families on one graph.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title: Exploring Families of Hyperbolas** **Description:** In this exercise, we will examine two families of hyperbolas: 1. The curves defined by \(xy = k\), where \(k\) is a real number. 2. The curves defined by \(x^2 - y^2 = m\), where \(m\) is a real number. **Objectives:** - **Orthogonality:** Demonstrate that these two families are orthogonal to each other at any intersection point of a curve from one family with a curve from the other family. - **Avoiding Degeneracy:** We will exclude the origin in our considerations, as it presents a point of degeneracy in the context of this exercise. - **Graphical Representation:** Sketch both families on a single graph for a visual understanding. Through this exploration, students will gain insight into the geometric properties and relationships between hyperbolic curves, particularly focusing on their orthogonal intersections and graphical representations.
Transcribed Image Text:**Title: Exploring Families of Hyperbolas** **Description:** In this exercise, we will examine two families of hyperbolas: 1. The curves defined by \(xy = k\), where \(k\) is a real number. 2. The curves defined by \(x^2 - y^2 = m\), where \(m\) is a real number. **Objectives:** - **Orthogonality:** Demonstrate that these two families are orthogonal to each other at any intersection point of a curve from one family with a curve from the other family. - **Avoiding Degeneracy:** We will exclude the origin in our considerations, as it presents a point of degeneracy in the context of this exercise. - **Graphical Representation:** Sketch both families on a single graph for a visual understanding. Through this exploration, students will gain insight into the geometric properties and relationships between hyperbolic curves, particularly focusing on their orthogonal intersections and graphical representations.
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