Math Question

I'm in IB math methods, working on a project and I can't seem to figure out what to do with this problem.

A radio transmitter sends signals to a railway which run along a straight track. When a set of coodinate axes is used to represent this system, the transmitter is at R(1,0), and the track T is a line w/ equation 2x+y=30, where the units are kilometers.
- Points A and be are the points of T that cross the x and y axis
- The engine receives the strongest signal at C, which is the point on T closest to R.
- P(x,y) is a general point on T

My problem is this:
let s(x) represent the distance between R(1,0) and P(x,y). Show that
s(x)= sq rt (5x^2-122x+901).

Help! I don't even know where to begin. I said that point C is
(15x-15, -60x + 900). Since RP is (x-1)i + (-2x+30)j, does the 901 in the thing I'm supposed to show have to do with... oh never mind I'm REALLY confused. Please help me. Thanks!

poof

First, all we have to do is show that s(x) = sqrt (5x^2 - 122x + 901). It seems like your making this out to be alot more complicated then it really is. Forget vectors or anything other then good old algebra for a minute.

Recall the distance formula: sqrt[ (x2-x1)^2 + (y2-y1)^2 ]

We plug in our values for (x1,y1) (x2,y2) from the points given.

Therefore:

s(x) = sqrt [ (x-1)^2 + y^2 ]

Now we need to change that y^2 into x..we are in luck because they have given us the equation of the line on which these points are located. Solve 2x + y = 30 for y... we get: y = 30 - 2x

Substitute this back into the equation we just obtained with the distance formula and multiply it out.

s(x) = sqrt [ (x-1)^2 + (30-2x)^2 ]

s(x) = sqrt [ x^2 - 2x +1 + 900 - 120x + 4(x)^2 ]

simplify:

s(x) = sqrt [ 5(x)^2 - 122x + 901 ]

There you have it.

Thanks!

Thank you so much! I can't believe how simple that was. I have such a huge problem when I am asked to prove things. I just never seem to find the way to do it.

Another Problem

In the above problem, they want me to draw an arc with the center R with a radius of 28 and then find the length of the portion of T that will be within the range of the arc. The drawing the arc was no problem, but I don't know how to find it.