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in Co-ordinate geometry by (15 points)
edited by

Minimum distance between the curves \( y^{2}=x-1 \) and \( x^{2}=y-1 \) is equal to

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1 Answer

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by (32.8k points)

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.

⇒ y2 = x − 1

Differentiating w.r.t x we get,

and the other point of tangency is again swapped to

Now use distance formula between them to get the minimum distance

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