You are now allowed to assume that the half-planes determined by the line with the equation ax+by +c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Usingthis, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given lineand that (s, t) and (u, v) are on the same side of the given line. en construct the parametrized linethrough (q, r) and (u, v). Consider the mappingλ γ7→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for everyλ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.) You are now allowed to assume that the half-planes determined by the line with the equation ax+by+c = 0 correspond to the points (x, y) so that ax+by+c<0 and ax+by+c>0, respectively. Usingthis, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given lineand that (s, t) and (u, v) are on the same side of the given line. Then construct the parametrized linethrough (q, r) and (u, v). Consider the mappingXa(q-qλ + uλ) + b(r = rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for everyA, y(x) > 0 or y(x) < 0, depending on which half-plane the points are on.)

To show that axiom B4(i) holds, we need to prove that for any two distinct points (q, r) and (s,…

Answered: ax + by + c > 0,

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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Question

You are now allowed to assume that the half-planes determined by the line with the equation ax+by +
c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Using
this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line
and that (s, t) and (u, v) are on the same side of the given line. en construct the parametrized line
through (q, r) and (u, v). Consider the mapping
λ γ
7
→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every
λ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.)