Question

what is the formula of the distance between a point and a line

Answer 1

d = sqrt((x1-x2)^2 + (y1-y2)^2

This is just pythagoras theorem, applied to a straight line.

Answer 2

hold on

Answer 3

you have a line, say
\[y=2x+1\] and a point not on the line, say (3,8) and you want the distance right?

Answer 4

yes

Answer 5

or do you have as specific one you would like to do?

Answer 6

i have a calculus journal i have to do and it asks me the formula for a distance between a point and a line

Answer 7

because this is not that trivial

Answer 8

ooooooooooooooh you want to use calculus ok

Answer 9

so "distance" of course means "shortest distance" you can do it without calculus or you can use calculus. now if you want some sort of formula i guess we should call the line
\[y=mx+b\] and the point
\[(x_1,y_1)\] or something like that

Answer 10

ok and how to find the distance between those two?

Answer 11

ok first of all use the distance to find the point on the line closest to the point
\[(x_1,y_1)\] the point is given and the line is given. the only variable is x and y

Answer 12

put
\[d^2=(x-x_1)^2+(y-y_1)^2\]

Answer 13

you want to find x and y that make this a minimum. x and y being on the line y = mx +b

Answer 14

you mean like in vector?

Answer 15

so replace y by mx + b in the formula to get
\[d^2(x)=(x-x_1)^2+(mx+b-y_1)^2\]

Answer 16

now you have a function of x, it is smallest where the derivative is 0, so take the derivative, set it = 0 and solve

Answer 17

remember that the variable is x,
\[x_1,y_1,b\] are constants

Answer 18

@imranmeah if there is a snap way let me know. i would make the slopes perpendicular, but we are supposed to use calculus

Answer 19

\[\frac{\left|P_x\times n\right|}{|n|}\]
this one?

Answer 20

thanks is there a specific formula. i googled it and i got a bunch of different formulas

Answer 21

not that one but it does have an absolute value in it

Answer 22

my understanding of the question was we have a line
\[y=mx+b\] and a point
\[(x_1,y_1)\] and we want to use calculus to find the distance between the point and the line

Answer 23

the derivative is
\[2(x-x_1)+2(mx+b-y_1)\] by the chain rule. set this equal to zero, solve for x. you answer will have an
\[x_1,y_1,m, b\] in it which is to be expected because you want a formula which will obviously depend on those 4 numbers

Answer 24

a quick bit of algebra tells me
\[x=\frac{x_1+y_1-b}{m+1}\]

Answer 25

now that is not the formula for the distance, that is the x - value of the point on the line that gives the distance. the y value is what you get if you replace x by this mess and the distnance is what you get from the distance formula

Answer 26

ok thanks