The smallest distance between the origin and a point on the parabola y = x^2 -5 can be expressed as "sqrt(a)/b," where "a" is not divisible by the square of any prime. Find a+b .
+128406 y = x^2 - 5
Let the point on the parabola be (x, x^2 - 5)
The distance, D, between this point and the origin can be represented by :
D = sqrt [ x^2 + ( x^2 - 5)^2 ]
D = sqrt [ x^2 + x^4 - 10x^2 + 25]
D = [ x^4 - 9x^2 + 25 ]^(1/2)
Take the derivative of this and set to 0
D' = [(1/2] [(x^4 - 9x^2 + 25 ]^(-1/2)* (4x^3 - 18x) = 0
This will equal 0 when (4x^3 - 18x) = 0 factor this
2x (2x^2 - 9) = 0
Set both factors to 0 and solve for x
2x = 0 2x^2 - 9 = 0
x = 0 2x^2 = 9
reject x^2 = 9/2
So....the distance, D, is
D = sqrt [ (9/2)^2 - 9(9/2) + 25 ] = sqrt [ 81/4 - 81/2 + 25 ] =
sqrt [ 81/ 4 - 162/4 + 100/4] =
sqrt [ 81 - 162 + 100] / 2 =
sqrt [ 19 ] /2
So....a + b = 19 + 2 = 21
