Question

I need some help....@Amistre,Satellite,Watchmath and whoever who can give a solution to my problem...
A(2;5) B(5;-1) C(x;y)
A,B and C are in the same line, 2|AC|=5|BC| find x+y=?

Answer 1

Hm. Is there an illustration that goes along with this?

Answer 2

Oh never mind. I didn't notice C(x;y)

Answer 3

kk :)

Answer 4

Ok. And just to double-check, |AC| is presumably the length of the A-C line segment?

Answer 5

we r supposed to work with vectors in here

Answer 6

Right, so the magnitude of the vector. Sounds good.

Answer 7

And B is between A and C

Answer 8

Ok, so really quick we can see that:

\[\begin{align}|AB| = \sqrt{(5 - 2)^2 + (-1 - 5)^2}\end{align}\]

We can see conclude that:

\[\begin{align}|AC| &= \sqrt{(x - 2)^2 + (y - 5)^2}\\
|BC| &= \sqrt{(x - 5)^2 + (y - -1)^2}\end{align}\]

Answer 9

So if 2|AC| = 5|BC| then:

\[\begin{align}
2\sqrt{(x-2)^2 + (y-5)^2} &= 5\sqrt{(x - 5)^2 + (y + 1)^2}\\
\left(2\sqrt{(x-2)^2 + (y-5)^2}\right)^2 &= \left(5\sqrt{(x - 5)^2 + (y + 1)^2}\right)^2
\end{align}\]

We can square both sides to make those ugly square roots go away:


\[4(x - 2)^2 + 4(y - 5)^2 = 25(x - 5)^2 + 25(y + 1)^2\]

Answer 10

won't u have -x^2-66x-12y-85?

Answer 11

or smth like this

Answer 12

oops w8 I've made a mistake :(

Answer 13

Then we do the annoying multiplication:

\[\begin{align}
4(x^2 - 4x + 4) + 4(y^2 - 10y + 25) &= 25(x^2 - 10x + 25) + 25(y^2 + 2y + 1)\\
4x^2 - 16x + 16 + 4y^2 - 40y + 100 &= 25x^2 - 150x + 625 + 25y^2 + 50y + 25\\
4x^2 - 16x + 4y^2 - 40y + 116 &= 25x^2 - 150x + 25y^2 + 50y + 650\\
-534 &= 21x^2 - 134x + 21y^2 + 10y\\
\end{align}\]

Now the question is how to get x + y from this.

Answer 14

pet22, post your question on the right and wait your turn. Please don't hijack other questions.

Answer 15

ok

Answer 16

If you repost it I'll get to it in a second.

Answer 17

I wonder if I took the wrong approach here :)

Answer 18

i got 566 not 534....
anyway i don't think i know how to solve this O.o

Answer 19

If it may help my teacher has expressed AC as AB+BC
so (AB+BC)/BC=5/2
AB/BC=5/2-1=3/2
BC=2AB/3
I got it.....

Answer 20

Thanks :):):)

Answer 21

Aha. Nice! Rock on :)