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Find the point on the line 7x+1y−5=0 which is closest to the point (−2,−6).

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  1. 7x+1y−5=0
    y=-7x+5
    so the distance minimum will be on a line thru the point (-2,6) and has a slope of 1/7

    y=1/7 x+b
    -2=1/7 (-2)+b
    b=-2+2/7

    now look at the lines to find the intersection.
    y=-7x+5
    y=x/7 -2+2/7
    set them equal
    -7x+5=x/7-2+2/7
    solve for x, then go back and solve for y. That (x,y) is the point on the original line which is closest to (-2,-6)

  2. The distance from (-2,-6) to the line 7x+y-5=0 is

    |7(-2)+1(-6)-5|/√(7^2+1^2) = 25/√50 = 5/√2

    so, we need to find (x,y) on the line such that

    (x+6)^2 + (y+2)^2 = 25/2
    (x+6)^2 + (11-7x)^2 = 25/2
    x = 3/2
    so, y=5-7x = -11/2
    So, (3/2,-11/2) is the closest point to (-2,-6)

    check: the slope of the line joining those two points is
    (1/2)/(7/2) = 1/7
    as shown above

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