CALC

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point on the hyperbola 9x^2−6y^2=10 closest to the point (0, 7).

y coordinate of each point= ?
positive x coordinate= ?
negative x coordinate= ?

  • CALC -

    let the point be P(a,b)
    slope of line form P to (0,7) = (b-7)/a

    differentiate:
    18x - 12y dy/dx = 0
    dy/dx = 18x/(12y) = 3x/(2y)
    at P, dy/dx = 3a/(2b)
    so the slope of the tangent at P is 3a/(2b)
    by basic geometry the slope of that tangent and the slope of the line to (0,7) to P must be negative reciprocals of each other.
    so (b-7)/a = -2b/(3a)
    -2ab = 3ab - 21a
    5ab = 21a
    b = 21/5

    sub into 9a^2 - 6b^2 = 10
    9a^2 - 6(441/25) = 10
    a = ± 4√181/15

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