SOLUTION: i have a question if my multiple choice answers for my question find the shortest distance between the parallel lines with equations 5x-12y+33=0 and 5x-12y-6=0 A.3 B.39 C.2

Algebra ->  Decimal-numbers -> SOLUTION: i have a question if my multiple choice answers for my question find the shortest distance between the parallel lines with equations 5x-12y+33=0 and 5x-12y-6=0 A.3 B.39 C.2      Log On


   

Question 147056: i have a question if my multiple choice answers for my question
find the shortest distance between the parallel lines with equations 5x-12y+33=0 and 5x-12y-6=0
A.3
B.39
C.27/5
D.27/13
E.N/a
how did u come up with the answer 3.25
Found 2 solutions by jim_thompson5910, edjones:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
5x-12y%2B33=0 Start with the first equation.


-12y%2B33=0-5x Subtract 5x from both sides.


-12y=0-5x-33 Subtract 33 from both sides.


-12y=-5x-33 Combine like terms on the right side.


y=%28-5x-33%29%2F%28-12%29 Divide both sides by -12 to isolate y.


y=%285%2F12%29x%2B11%2F4 Simplify.



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5x-12y-6=0 Now move onto the given equation.


-12y-6=0-5x Subtract 5x from both sides.


-12y=0-5x%2B6 Add 6 to both sides.


-12y=-5x%2B6 Combine like terms on the right side.


y=%28-5x%2B6%29%2F%28-12%29 Divide both sides by -12 to isolate y.


y=%285%2F12%29x-1%2F2 Simplify.



So when we graph the two equations, we get


Graph of y=%285%2F12%29x%2B11%2F4 (red) and y=%285%2F12%29x-1%2F2 (green)


Notice how the slope of the two equations is m=5%2F12. So the perpendicular slope is -12%2F5. Now simply draw any perpendicular line to the equations. I'm going to graph y=-%2812%2F5%29x




Graph of y=%285%2F12%29x%2B11%2F4 (red) , y=%285%2F12%29x-1%2F2 (green), and the perpendicular line y=-%2812%2F5%29x (blue)



Now use any method to find the two intersections. I used a graphing utility to get the two intersections at the points and



Now let's find the distance between the two points. This distance will be the same as the shortest distance between the two lines




d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%28-165%2F169-30%2F169%29%5E2%2B%28396%2F169--72%2F169%29%5E2%29 Plug in x%5B1%5D=-165%2F169, x%5B2%5D=30%2F169, y%5B1%5D=396%2F169, and y%5B2%5D=-72%2F169. These are the coordinates of the intersections.


d=sqrt%28%28-15%2F13%29%5E2%2B%28396%2F169--72%2F169%29%5E2%29 Subtract 30%2F169 from -165%2F169 to get -15%2F13.


d=sqrt%28%28-15%2F13%29%5E2%2B%2836%2F13%29%5E2%29 Subtract -72%2F169 from 396%2F169 to get 36%2F13.


d=sqrt%28225%2F169%2B%2836%2F13%29%5E2%29 Square -15%2F13 to get 225%2F169.


d=sqrt%28225%2F169%2B1296%2F169%29 Square 36%2F13 to get 1296%2F169.


d=sqrt%289%29 Add 225%2F169 to 1296%2F169 to get 9.


d=3 Take the square root of 9 to get 3.


So our answer is d=3


So the distance between the two points is 3 units.


So this means that the shortest distance between the two lines is also 3 units.

So the answer choice is A)

Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
5x-12y+33=0
5x-12y-6=0
.
5x-12y+33=0
-12y=-5x-33
y=5x/12 + 33/12
.
5x-12y-6=0
-12y=-5x+6
y=5x/12 - 1/2
.
The distance between the parallel lines is |33/12 + 1/2|=3.25
This is easier to see if we change the slope from 5x/12 (1st graph) to 0x/12 (2nd graph)

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Ed
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