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consider the line y=6x-k and the parabola y=x^2

i) for what value of k is the line y=6x-k a tangent to the parabola y=x^2 ?

ii) the line y=6x-k intersects the parabola in two distinct places. what is the largest integer value that k can take ?

  • maths -

    i. at x=3, the slope of the parabola is 6, so at 3, the parabola has a y value of 9, which means if the line y=6x+k is =9, then k=....

  • maths -

    i)
    using Calculus, this is easy
    if y = x^2
    dy/dx = 2x
    but the slope of the line is 6
    so 2x = 6
    x = 3
    if x=3 then y=9 from the parabola
    so the point (3,9) must also be on the line
    9 = 6(3)-k
    k = 9

    ii) let's intersect
    y = x^2 and y = 6x-k
    then x^2 - 6x + k = 0

    to have 2 distinct roots, the discriminat must be positive
    so b^2 - 4ac > 0
    36 - 4(k) > 0
    k < 9

    since k=9 produces the tangent, and you asked for the largest integer that k could have,
    k = 8
    (strange question!)

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