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Calculus

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Find the point on the line -6 x + 5 y - 1 =0 which is closest to the point ( -4, 1 )

  • Calculus -

    Let the point be P(x,y) , point (-4,1) be A
    Since the shortest distance must be where AP is perpendicular to the given line.
    slope of given line = 6/5
    so slope of AP is - 5/6
    equation of AP:
    y-1 = (-5/6)(x+4)
    times 6
    6y - 6 = -5x- 20
    5x + 6y = -14 , #2
    6x - 5y = -1 , #1, the given line

    #1 times 6 ---->36x - 30y = -6
    #2 times 5 ----> 25x + 30y = -70
    add them:
    61x = -76
    x = -76/61
    back into #1
    6(-76/61) - 5y = -1
    -5y = -1 + 456/61
    -5y = 395/61
    y = - 79/61

    The closest point is (-76/61 , -79/61)


    check my arithmetic, expected "nicer " numbers.

  • Calculus -

    just as a check, the distance from point P(h,k) to the line ax+by+c=0 is

    |ah+bk+c|/√(a^2+b^2)

    In this case, that is

    |24+5+1|/√(36+25) = 30/√61

    Hmmm. What went wrong above?

    Oops. Forgot to multiply -6 by 6

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