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Question

Minimum distance between the curves y2=x1 and x2=y1 is a2b, where a,b are co-prime numbers, then the value of a+b is

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Solution

Given : y2=x1 and x2=y1
By observation, if we swap x by y in y2=x1, then we get x2=y1
So, they are symmetric about y=x
Finding the equation of tangents on both the curve which is parallel to y=x
Now for y2=x1,
2yy=1y=12y=1y=12x=54xy34=0
Now for x2=y1,
y=2x2x=1x=12y=54xy+34=0

Distance between parallel lines
=∣ ∣ ∣ ∣343412+12∣ ∣ ∣ ∣=324

Hence, a+b=7

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