The distance of point (-1, 4) and the line 4x – 2y = 4 is ___.

Question

anonymous · Answer

8/sqart 5

moongazer · Answer

how did you do it?

anonymous · Answer

$$8/\sqrt{5}$$

moongazer · Answer

how?

moongazer · Answer

Please explain.

anonymous · Answer

wait, wait

anonymous · Answer

4x-2y-4=0
put the values of x and y in the LHS
then take the modulus of that value and divide it by sqrt of sum of the squares of coeffients of x and y. i think, u know the proof why we do that

anonymous · Answer

sorry, co-efficients ^^

moongazer · Answer

Sorry but I didn't get it.

amistre64 · Answer

since the distance from a point to a line is along a perpendiular line; if we can construct a perpendicular line equation that goes thru the given point we can use a system of equations to determine the point where they meet.  then use the distance formula to determine the distance from the meeting point and the given point.

amistre64 · Answer

|dw:1326977726363:dw|

amistre64 · Answer

one trick to a perp line is to swap coeffs and negate one of them:

4x – 2y = 4

perp line:  2x + 4y = n

calibrate with the given point (-1,4):  2(-1) + 4(4) = n
                                                            -2   +  16  = n = 14

perp line:  2x + 4y = 14

amistre64 · Answer

system of equations gives us:

4x - 2y  =  4
2x + 4y = 14  ; *-2


 4x - 2y  =  4
-4x -8y = -28
------------
     -10y = -24

          y = 24/10 = 12/5

..............................................

4x - 2y  =  4   ;*2
2x + 4y = 14



8x - 4y  =  8
2x + 4y = 14
------------
10x        = 22

           x = 22/10 = 11/5

................................

we want the distance from (-1,4) to (12/5, 11/5)   if i did that right

moongazer · Answer

I think this is the explanation that I am looking for.

moongazer · Answer

could I use the formula d=(/Ax1+By1+C/)/sqrtof(a^2+B^2)
I think this is the formula for this. :)

amistre64 · Answer

$$dist.=\sqrt{a^2+b^2}$$
is a compact form of the distance formula yes; personally I just subtract the points, square them, add them and sqrt them

amistre64 · Answer

( 12/5,  11/5) 
- (  -5/5, 20/5) 
---------------
   (17/5 ,  -9/5) ^2

    (289+81)/25

     sqrt(370)/5

amistre64 · Answer

that should simplify to 8/sqrt(5)

amistre64 · Answer

but then some texts hate a sqrt in the denom; so maybe we rewrite it:  $cfrac{8}{5}\sqrt{5}$

amistre64 · Answer

well that dint format good on my screen lol

amistre64 · Answer

$$\frac{8\sqrt{5}}{5}$$
or
$$\frac{8}{5}\sqrt{5}$$

moongazer · Answer

$$ d= (\left| Ax _{1}+Bx _{2}+C \right|)/\sqrt{A ^{2}+B ^{2}}$$
This is what my teacher taught us :)

amistre64 · Answer

that might work :)  ive never quite seen it like that tho.