Have a look at this graph, Melody.
https://www.desmos.com/calculator/blatwtgdae
I have graphed the tangent line to the curve at (1/√2, 3/2) and a line perpendicular to this one that goes through the same point. And the perpendicular line also goes through (0,2). And the shortest distance between a line and a point is a perpendicular line to the given line passing through that point....
{The same argument can be made for the other point (-1/√2, 3/2)...}
Call the point(s) on the parabola that are closest to (0,2), (x, x^2 + 1)
And the distance between these two is given by.....
d = [(x-0)^2 + (x^2 + 1 - 2)^2 ]^(1/2)
d = [(x^2 + (x^2 -1)^2]^(1/2)
d = [(x^2 + x^4 - 2x^2 + 1]^(1/2)
d = [x^4 - x^2 + 1]^(1/2)
We wish to minimize this.....take the derivative....we have
d' = (1/2)[x^4 - x^2 + 1]^(-1/2)*(4x^3 - 2x) set to 0
This will be 0 when the numerator= 0 so we have
(4x^3 - 2x) = 0 factor
2x(2x^2 -1 ) = 0 set each factor to 0
So, either x = 0 or x = ±√(1/2)
And when x = 0 y =(0)^2 + 1 = 1 so (0,1) is one possible point
And when x = ±√(1/2), y = (±√(1/2))^2 + 1 = 3/2
So the other two possible points are (-√(1/2), 3/2) and (√(1/2), 3/2)
Check the distance between(0, 2) and (0, 1) =1
Check the distance between (0,2) and (√(1/2), 3/2)
d = [(√(1/2))^2 + (2 -3/2)^2]^.5 = [(1/2) + (1/2)^2]^.5 = [(1/2 + 1/4]^.5 = (3/4)^.5 = √3/2 ... and this distance will be the same between (0,2) and (-√(1/2), 3/2)
And these two distances are less than the distance between (0,2) and (0,1)
So, the two points on the parabola y = x^2 + 1 that are closest to (0,2) are (±√(1/2), 3/2)
Thanks Chris,
I'm all confused, algegraically what you say seems correct but look at the pic.
I think the answer is (0,0), (1,2) and (-1,2)My algebra is slightly diff from your but I get the same answer - i am sure it is wrong.
It's doing my head in - I am going to take a break. :))
Have a look at this graph, Melody.
https://www.desmos.com/calculator/blatwtgdae
I have graphed the tangent line to the curve at (1/√2, 3/2) and a line perpendicular to this one that goes through the same point. And the perpendicular line also goes through (0,2). And the shortest distance between a line and a point is a perpendicular line to the given line passing through that point....
{The same argument can be made for the other point (-1/√2, 3/2)...}