Let L be a tangent line to the hyperbola x y = 2 at x = 9 . Find the area of the triangle bounded by L and the coordinate axes. ( Give your

Question

Let L be a tangent line to the hyperbola x y = 2 at x = 9 . Find the area of the triangle bounded by L and the coordinate axes. ( Give your answer as a whole or exact number.)

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Allison 2 days 2021-10-11T15:39:33+00:00 1 Answer 0

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    2021-10-11T15:40:46+00:00

    Answer:

    A = 4

    Step-by-step explanation:

    The equation of the slope of the tangent line L is obtained by deriving the equation of the hyperbola:

    y = \frac{2}{x}

    y'=-2\cdot x^{-2}

    The numerical value of the slope is:

    y' = -2 \cdot (9)^{-2}\\y' = -\frac{2}{81}

    The component of the y-axis is:

    y = \frac{2}{9}

    Now, the tangent line has the following mathematical model:

    y = m \cdot x + b

    The value of the intercept is found by isolating it within the equation and replacing all known variables:

    b = y - m \cdot x

    b = \frac{2}{9}-(-\frac{2}{81} )\cdot (9)\\b = \frac{4}{9}

    Thus, the tangent line is:

    y = -\frac{2}{81}\cdot x + \frac{4}{9}

    The vertical distance between a point of the tangent line and the origin is given by the intercept.

    d_{y} = \frac{4}{9}

    In order to find horizontal distance between a point of the tangent line and the origin, let equalize y to zero and clear x:

    -\frac{2}{81}\cdot x + \frac{4}{9}=0

    -\frac{2}{9}\cdot x + 4 = 0

    x = 18

    d_{x} = 18

    The area of the triangle is computed by this formula:

    A = \frac{1}{2}\cdot d_{x}\cdot d_{y}

    A = \frac{1}{2}\cdot (18)\cdot (\frac{4}{9} )

    A = 4

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