Distances Between Curves

★ 7/10ExtremacalculusReflectionanalytic geometry

Problem

For curves C1,C2C_1, C_2, define their distance d(C1,C2)d(C_1, C_2) as the minimum of ∣PQ∣|PQ| over P∈C1P \in C_1, Q∈C2Q \in C_2.

1. Find d(C1,C2)d(C_1, C_2) for

C1 ⁣:x2+y2=2,C2 ⁣:(x−3)2+(y−3)2=2.C_1 \colon x^2 + y^2 = 2, \qquad C_2 \colon (x-3)^2 + (y-3)^2 = 2.
  1. Find d(C3,C4)d(C_3, C_4) for
C3 ⁣:ex−2y=0,C4 ⁣:ln⁡x+ln⁡2=y.C_3 \colon e^x - 2y = 0, \qquad C_4 \colon \ln x + \ln 2 = y.

Answer

Solution

Difficulty7/10
TopicsExtrema, calculus, Reflection, analytic geometry

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