Question

P is 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

Answer 1

The minimum distance is the perpendicular distance. You need to select an arbitrary point on the line, (x,y), say, and establish the distance from the origin to the line using the distance formula. The distance here is,\[d^2=(x-0)^2+(y-0)^2=x^2+y^2\]You then need to minimize this function d^2 subject to the constraint, \[2x+y-10=0\]You can use implicit differentiation or just straight substitution. If we substitute, the y-values the distance function can take will be related to the x-values by the line:\[y=10-2x\]You can substitute this in for y in the distance function and take the derivative,\[d=\sqrt{x^2+(10-2x)^2}\]Expand and simplify the radicand (stuff under the square root) before taking the derivative fo d with respect to x.

Answer 2

\[d'=\frac{1}{2}(5x^2-40x+100)^{-1/2}(10x-40)\]Setting the derivative to zero to find optimal x,\[d'=0 \rightarrow 10x-40=0 \rightarrow x=4\]This will be the x-value on the line such that the distance between the origin and line will be EITHER a maximum or minimum (technically, it should be checked afterward). For x = 4, the corresponding y-value is found from the equation of the line (since we need the corresponding y-value on the line for this x-value). Then y = 10 - 2(4) = 2.

So the point, P, is (4,2).

Answer 3

thanks man, i differentiated twice, silly me -.-

Answer 4

np :)