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The shortest distance between the parabolas y^2 =x-1 and x^2=y-1 is?

The shortest distance between the parabolas y^2 =x-1 and x^2=y-1 is?

Grade:12

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
512-739_2016-01-17-131303.jpg
As shown in the figure due to the symmetry in parabolas the shortest distance AB will be perpendicular to the line x=y
so we just need to find the coordinates of A
tangent at A will be parallel to x=y => have slope 1
For finding that we differentiate P1, and equate the gradient as 1
\frac{dy}{dx} = 2x = 1
Thus coordinate of A = (0.5,1.25)
being reflexive about x=y we can say that coordinate of B = (1.25,0.5)
Hence the shortest distance
AB = \sqrt{0.75^2 + 0.75^2} = 0.75\sqrt{2}

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