Prove that the curves x = y² and xy = k cut at right angles* if 8k² = 1.

Question

Prove that the curves x = y² and xy = k cut at right angles* if 8k² = 1.

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Iris 2 weeks 2021-10-04T01:01:24+00:00 1 Answer 0 views 0

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    2021-10-04T01:02:48+00:00

    Answer:

    Given curves are x = y²——-( 1 ) ,

    xy = k ———–( 2 )

    => x = k/y

    x = y² => k/y = y²

    k = y³

    y = k⅓ [ put in ( 2 )

    x = k/y

    x = k/k⅓

    hence, the point of intersection of of curves are (k⅔ , k⅓)

    NOW,

    we know slope of tangent = dy/dx .

    for x = y²

    2y*dy/dx = 1

    dy/dx = 1/2y——-( 3 )

    and slope of tangent at (k⅔ , k⅓) is

    for xy = k

    x*dy/dx + y = 0

    dy/dx = -y/x

    the slope of tangent at (k⅔ , k⅓) is

    we know if two lines are perpendicular then product of there slope = -1

    now,

    (slope of tangent at curve x = y²) × (slope of tangent at curve xy = k) = -1

    on cubing both sides, we get,

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