calculus

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Determine the coordinates of the point of intersection of the two perpendicular lines that intersect on the y-axis and are both tangent to the parabola given below.
y = 3x2

  • calculus -

    let y=mx+b be the first line
    and y=1/m x+ d be the second line.

    If y=3x^2 is tangent, then m=3, or m=1/3
    .

    y=3x^2=mx+b so m=3, b=0

    y= 1/3 x +d, so m=1/3, d=0

  • calculus -

    oops. one of them has to be negative, so y=-1/3 x+ d, m=-1/3, d=0

  • calculus -

    Since the two tangents are mutually perpendicular and intersect on the y-axis they have gradients 1 and -1
    The gradient function is y = 6x
    Hence x = 1/6, -1/6
    So y = 1/12 both times
    So the coordinate of the intercept can be calculated using the equation of the tangents:
    Equate y - 1/12 = x - 1/6
    and y - 1/12 = -1(x + 1/6)
    BUT you do not even have to do this!
    Re-write one of the equations in the form
    y = mx + c to get the y-intercept.
    The coordinate required is
    (0, -1/12)

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