Question

consider points A(-3,5) and point B(x,2x). Find point B so that the distance between A and B is exactly √37.

Answer 1

Do you know the distance formula between 2 points... or the Pythagorean theorem for right triangles?

Answer 2

i tried the distance formula and set it equal to √37

Answer 3

that's a good approach... how did you set up the distance formula? Maybe you just made a small error...

Answer 4

i got down to 4x^2-13x-53=37

Answer 5

i set it up like √37= √(-3x-x)^2-(5-2x)^2

Answer 6

That first parenthesis should be (-3-x)^2 not (-3x-x)^2

Answer 7

yea sorry that was just a typo

Answer 8

stuff inside the square root:
= (-3-x)^2 - (5-2x)^2
= 9-6x+x^2 - 25 -20x + 4x^2
= 5x^2 -26x -16

Answer 9

shouldn't the -6x on the second line be +6x

Answer 10

yes, you're right.. .good catch, sorry...

Answer 11

so it should be 5x^2 -14x -16 ?

Answer 12

yes. i found my mistake of not having the single "x" squared. i forgot to square it and left it only as x

Answer 13

ok :) It's a pain when you make a small error like that... takes awhile to find it sometimes.

Did you already get a solution for x?

Answer 14

no

Answer 15

did you find an answer i am having trouble

Answer 16

Sorry for the delay... let me work it through from the beginning to make sure I don't keep injecting dumb errors due to working to quickly

Answer 17

no problem. Thanks

Answer 18

i got down to 5x^2-14x-53. and i was stuck. Is this atleast right?

Answer 19

Sqrt(37) = Distance between points = \[\sqrt{5x^2 - 14x + 34}\]

Answer 20

how did you get +34?

Answer 21

I didn't, yet... but that should be the expression for distance between the 2 points. I was having browser issues... didn't want to hold you up if you were ahead of me.

Answer 22

oo okay

Answer 23

(x- (-3))^2 = (x+3)^2 = x^2 + 6x +9

(2x-5)^2 = 4x^2 -20x + 25

inside sqrt: (x^2 + 6x +9) + (4x^2 -20x + 25)
= 5x^2 -14x + 34

all that is = sqrt(37), so...

sqrt(37) = sqrt( 5x^2 -14x +34)

0 = 5x^2 -14x +34 -37 = 5x^2 - 14x -3

0 = (5x -1)(x - 3)

x = -1/5 OR x = 3

Answer 24

Yea that is what I just got too
Thanks

Answer 25

thanks for your patience :) Looks straightforward to just use "distance formula", but it's tricky trying not to make algebra errors along the way.

Answer 26

tell me about it

Answer 27

one last thing

Answer 28

to answer the question would you plug in both points to see which one works

Answer 29

?

Answer 30

yeah, I was thinking about that too...

My instinct tells me that since both are valid solutions for x, there are in fact 2 points (x,2x) in which the distance from the other point is sqrt(37).

Answer 31

You could confirm by substituting both, but they should BOTH work.

Answer 32

so my answer would be both... x=3 AND -1/5?

Answer 33

What exactly does this mean?

Answer 34

it means two different points satisfy these two constraints:

Constraint 1: solution point has distance sqrt(37) from the point (-3,5)

Constraint 2: y value of the point = twice x value of the point i.e. (x,2x)

Answer 35

sorry that was a typo

Answer 36

my answer for point be would b both (3,6) and (-1/5, -2/5)??

Answer 37

yes, that's it.

It seems weird, I guess, but there are really two valid solutions for the point described in the problem... and if my drawing had been more to scale, you would see it easier there.

Answer 38

ok Thank you

Answer 39

glad to help :)