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. For a fixed AER, define the curve CA in R² by the equation y² − 2xy = x¹ + Ax + 4. You can view this family of curves on Desmos. Note this demo will not help you justify your answers below. (a) Prove that if A-6 and A6, then CA is a regular curve. You may use Wolfram Alpha to solve a 1-variable quartic equation.

. For a fixed AER, define the curve CA in R² by the equation y² − 2xy = x¹ + Ax + 4. You can view this family of curves on Desmos. Note this demo will not help you justify your answers below. (a) Prove that if A-6 and A6, then CA is a regular curve. You may use Wolfram Alpha to solve a 1-variable quartic equation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. For a fixed A € R, define the curve CA in R2 by the equation y² 2xy = x² + Ax + 4. You can view this family of curves on Desmos. Note this demo will not help you justify your answers below. (a) Prove that if A ‡ -6 and A ‡ 6, then C₁ is a regular curve. You may use WolframAlpha to solve a 1-variable quartic equation. (b) This family of curves can instead be viewed as the z-slices of the surface S in R³ defined by the equation y² - 2xy = x² +xz+4 Use the implicit function theorem to show that this equation defines a 2-dimensional regular surface (c) The curves C6 and C_6 are regular at each point except (-1,−1) and (1,1) respectively. These points where C6 and C-6 fail to be regular are called singularities. View this surface and family of curves on Math3d. (Google Chrome is the most stable browser for Math3D.) Informally explain how the shape of the surface appears to relate to the singularities of C6 and C-6. Your explanation must discuss both the surface and the curves.
Transcribed Image Text:5. For a fixed A € R, define the curve CA in R2 by the equation y² 2xy = x² + Ax + 4. You can view this family of curves on Desmos. Note this demo will not help you justify your answers below. (a) Prove that if A ‡ -6 and A ‡ 6, then C₁ is a regular curve. You may use WolframAlpha to solve a 1-variable quartic equation. (b) This family of curves can instead be viewed as the z-slices of the surface S in R³ defined by the equation y² - 2xy = x² +xz+4 Use the implicit function theorem to show that this equation defines a 2-dimensional regular surface (c) The curves C6 and C_6 are regular at each point except (-1,−1) and (1,1) respectively. These points where C6 and C-6 fail to be regular are called singularities. View this surface and family of curves on Math3d. (Google Chrome is the most stable browser for Math3D.) Informally explain how the shape of the surface appears to relate to the singularities of C6 and C-6. Your explanation must discuss both the surface and the curves.
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