Question

Determine the point on the line y=2x+3 so that the distance between the line and the point (1,2) is a minimum.

Answer 1

@ganeshie8

Answer 2

geometrically, we can find the point easily by solving system of equations :
1) y=2x+3
2) y = -1/2x + 5/2

we want to work it usign calculus, right ?

Answer 3

yes but how did you get that second equation?

Answer 4

geometricly ,the minimum distance is the perpendiculer one

Answer 5

shortest distance between point and a line = `perpendicular` distance from the point to the line

Answer 6

lets work ut using calculus by optimizing the `distance` expression

Answer 7

oh ok

Answer 8

any point on the given line = (x, y) = (x, 2x+3)
given point : (1, 2)

Answer 9

optimizing distance(d) is same as optimizing its square(d^2) :

\[\large (x-1)^2 + (2x+1)^2\]

Answer 10

optimize ^^

Answer 11

where did 2x+1 come from?

Answer 12

apply distance formula for the points :
(x, 2x+3)
(1, 2)