The points of contact of the tangents drawn from the origin to the curve y=sin⁡x lie on the curve

The points of contact of the tangents drawn from the origin to the curve y=sinx lie on the curve

  1. A

    x2y2=xy

  2. B

    x2+y2=x2y2

  3. C

    x2y2=x2y2

  4. D

    none of these 

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    Solution:

    Let (h, k) be a point of contact of the tangents drawn from the origin to y = sin x. Then, (h, k) lies on y = sin x.

     k=sin h                                                       …(i)

    Now, 

    y=sinxdydx=cosxdydx(h,k)=cosh

    The equation of the tangent at (h, k) is

    yk=(cosh)(xh)

    It passes through (0, 0).

     k=hcoshkh=cosh                       ..(ii)

    From (i) and (ii), we get 

    k2+k2h2=1                                   [Squaring and adding] 

      h2k2=k2h2    

    Hence, the locus of (h, k) is x2y2=x2y2.

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