Go for it!
P = ( a − b ) 2 + ( 8 − b 2 − a 2 5 ) 2
P is defined above for some real numbers a and b . Find the minimum value of P .
The answer is 18.
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2 solutions
Same approach!
But after finding a=5 and b = 2, I found the distance between the two points and got ( 1 8 ) and my immediate reaction was reveal solution and then I realized that we had to find square of the distance(--facepalm--)
It happens with everyone.
By AM - GM inequality 2 ( a − b ) 2 + ( 8 − b 2 − a 2 5 ) 2 ≥ ( ( a − b ) ( 8 − b 2 − a 2 5 ) ) 2 and the inequality is only one equality when ( a − b ) 2 = ( 8 − b 2 − a 2 5 ) 2 ⇒ a − b = ± ( 8 − b 2 − a 2 5 ) ⇒ we have two equations, let's choose one: a − b = − 8 − b 2 + a 2 5 .This implies that ( a , b ) = ( 5 , 2 ) is a solution and then we can get a minimum for P = 1 8 ....
I have a geometric solution.P represents the square of the distance between any 2 points on the curves xy=25 and the portion of the circle x 2 + y 2 = 8 above the X-axis respectively.Hence,we want the square of shortest distance between the two curves.Symmetrically, the shortest distance is along y=x.The square of the shortest distance can then be computed,which comes out to be 18.