Let A be the area between co-ordinate axis, y2=x−1,x2=y−1 and the line which makes the shortest distance between two parabolas and

Let A be the area between co-ordinate axis, y2=x−1,x2=y−1 and the line which makes the shortest distance between two parabolas and A be the area between x=0,x2=y−1,x=y and the shortest distance between y2=x−1andx2=y−1,

1 thought on “Let A be the area between co-ordinate axis, y2=x−1,x2=y−1 and the line which makes the shortest distance between two parabolas and”

  1. Answer:

    Notice that these are inverse functions of each other, you can swap x,y to get to the second parabola.

    They are mirror images with respect to line x=y.

    Required point should have this slope y′=1 for its tangent at point of tangency at ends of common normal.

    Take the parabola with its symmetry axis coinciding with axis.

    ⇒y

    2

    =x−1

    Differentiating w.r.t x we get,

    ⇒2yy′=1

    ⇒2y=1

    ⇒y=

    2

    1

    ⇒(

    2

    1

    )

    2

    =x−1

    ⇒x=

    4

    5

    Hence the x,y coordinates are

    ⇒(x,y)=(

    4

    5

    ,

    2

    1

    )=(x

    1

    ,y

    1

    ) (say)

    and the other point of tangency is again swapped to

    ⇒(x

    2

    ,y

    2

    )=(

    2

    1

    ,

    4

    5

    )

    Now use distance formula between them to get the minimum distance

    ⇒d=

    (

    4

    5

    2

    1

    )

    2

    +(

    2

    1

    4

    5

    )

    2

    ⇒d=

    (

    4

    3

    )

    2

    +(

    4

    −3

    )

    2

    ⇒d=

    4

    3

    2

Leave a Comment