Consider line segments which are tangent to a point on the right half (x>0) of the curve y = x^2 + 1 and connect the tangent point to the x-axis. If the tangent point is close to the y-axis, the line segment is long. If the tangent point is far from the y-axis, the line segment is also very long. Which tangent point has the shortest line segment?

(Suppose C is a positive number. What point on the curve has first coordinate equal to C? What is the slope of the tangent line at that point? Find the x-intercept of the resulting line. Compute the distance between the point on the curve and the x-intecept, and find the minimum of the square of that distance (minimizing the square of a positive quantity gets the same answer as minimizing the quantity, and here we get rid of a square root).)

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  1. following the hints suggested:
    let the point be (c,c^2 + 1)
    dy/dx = 2x
    so at (c,c^2+1) the slope of the tangent is 2c

    let the tangent equation be y = mx + b
    y = 2cx + b
    for our point,
    c^2 + 1 = 2c(c) + b
    b = 1-c^2

    so the tangent equation is
    y = 2cx + 1-c^2
    at the x-intercept,
    0 = 2cx + 1-c^2
    x = (c^2 - 1)/(2c)

    then using the distance formula

    D^2 = (c^2+1)^2 + (c - (c^2 - 1)/(2c))^2

    Ok, I will now expand this. At first I thought to differenctiate using quotient rule for the last term, but it looked rather messy

    D^2 = c^4 + 2c^2 + 1 + c^2 - c^2 + 1 + (c^4-2c^2+1)/(4c^2)
    = c^4 + 2c^2 + 2 + (1/4)c^2 - 1/2 + (1/4)c^-2

    2D(dD/dc) = 4c^3 + 4c + c/2 - 1/(2c^3) = 0 for a max/min of D
    8c^6 + 9c^4 - 1 = 0

    getting really messy....
    let a = c^2
    solve 8a^3 + 9a^2 - 1 = 0
    a=-1 works !!!!!!
    (a+1)(8a^2 + a - 1) = 0
    if a=-1, c^=-1 ---> no solution
    8a^2 + a - 1 = 0
    a = (-1 ± √33)/16 = .2965 or a negative
    c^2 = .2965
    c = .5145

    Please, please check my arithmetic and algebra, the method is right!

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  2. Thanks so much, but then do you plug in c back into the first point and tangent line and x-intercept to get the x and y coordinates and tangent line equation and x-intercept, respectively?

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  3. Can you explain to me how you found out the distance formula. Because how did you find out D^2 = c^4 + 2c^2 + 1 + c^2 - c^2 + 1 + (c^4-2c^2+1)/(4c^2)? I understand the first part with c^4 + 2c^2 + 1, but I don't understand how you got the second part.

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  4. And isn't (c^4-2c^2+1)/(4c^2) supposed to be (c^4+2c^2+1)/(4c^2)?

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