SOLUTION: The distance between the points A and B is \sqrt{34}. If A = (a,4) and B = (-4,5), then find the sum of all possible values of a.

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Question 1209010: The distance between the points A and B is \sqrt{34}. If A = (a,4) and B = (-4,5), then find the sum of all possible values of a.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
The distance between the points A and B is \sqrt{34}. If A = (a,4) and B = (-4,5),
then find the sum of all possible values of a.
~~~~~~~~~~~~~~~~~~~~~~~ 

From the condition, we have this distance formula

    34 = %28a-%28-4%29%29%5E2 + %284-5%29%5E2,

or

    34 = %28a%2B4%29%5E2 + 1%5E2,

    34 = %28a%2B4%29%5E2 + 1,

    33 = %28a%2B4%29%5E2,

    a + 4 = +/- sqrt%2833%29,

    a%5B1%2C2%5D = -4 +- sqrt%2833%29.


Thus  a%5B1%5D + a%5B2%5D = -4 + sqrt%2833%29 -4 - sqrt%2833%29 = -8.


ANSWER.  The sum of all possible values of "a" is -8.

Solved.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

d = distance from A to B
d+=+sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Distance formula

sqrt%2834%29+=+sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Plug in the given distance

sqrt%2834%29+=+sqrt%28%28a-%28-4%29%29%5E2%2B%284-5%29%5E2%29 Plug in (x1,y1) = (a,4) and (x2,y2) = (-4,5)

sqrt%2834%29+=+sqrt%28%28a%2B4%29%5E2%2B%28-1%29%5E2%29

sqrt%2834%29+=+sqrt%28%28a%2B4%29%5E2%2B1%29

34+=+%28a%2B4%29%5E2%2B1 Square both sides

Let's replace 'a' with x to get 34+=+%28x%2B4%29%5E2%2B1

That rearranges to x%5E2%2B8x-17=0 which I'll let the student handle the scratch work.

Since the leading coefficient is 1, the roots p and q add to the negative of the x coefficient (refer to Vieta's Formulas; specifically the quadratic version)

Therefore we determine p+q = -8 which is the final answer