calculus

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Find the point on the line 4x+2y+5=0 which is closest to the point (-2,-1)

  • calculus -

    without calculus, the distance from the point to the line is

    |4(-2)+2(-1)+5|/√(4^2+2^2) = √5/2

    Using calculus, we have the distance z from (-2,-1) to (x,y) is

    z^2 = (x+2)^2 + (y+1)^2
    But y = -(4x+5)/2, so
    z^2 = (x+2)^2 + ((-(4x+5)/2)+1)^2
    z^2 = 5x^2 + 10x + 25/4

    z = √(5x^2 + 10x + 25/4)
    dz/dx = 10(x+1)/√(5x^2+10x+25/4)

    dz/dx=0 at x=-1
    z(-1) = √5/2

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