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

Question

precal · Answer

@ganeshie8

ganeshie8 · Answer

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 ?

precal · Answer

yes but how did you get that second equation?

ganeshie8 · Answer

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ikram002p · Answer

geometricly ,the  minimum distance is the perpendiculer one

ganeshie8 · Answer

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

ganeshie8 · Answer

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

precal · Answer

oh ok

ganeshie8 · Answer

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

ganeshie8 · Answer

optimizing distance(d) is same as optimizing its square(d^2) :

$$\large    (x-1)^2 + (2x+1)^2$$

ganeshie8 · Answer

optimize ^^

precal · Answer

where did 2x+1 come from?

ganeshie8 · Answer

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