Pis the point on the line 2x+y-10=0 such that the length of op, the line segment from the origin o to p is a minimum. find the coordinates of p and this minimum length

Coolio

alright, so the disatnce formula is


solve for y in the equation given on top
2x+y-10=0
y=10-2x
ok
so (x1,y1)=(0,0), the first point
(x2,y2)=(x,10-2x)
so subsitute to get





find the miniumum
we only need to minimize what is under the square root so
take the derivitive of 5x²-40x+100
that is 10x-40
it is 0 at x=4
when x<4, the derivitive is negative
when x>4, the derivitive is positve
goes from negative to positive
so it is a minimum

at x=4
find y
y=10-2x
y=10-2(4)
y=10-8
y=2

(4,2)
find min distance








the point, P, is (4,2) and the minimum distance is 2√5