Find a point on the parabola with given point and distance

theintarnets

1. The problem statement, all variables and given/known data

Find a point on the parabola y = x^2 whose distance from (18,0) is 4*sqrt(17)


2. Relevant equations

I'm guessing I need to use the distance formula, but I'm not entirely sure.
D = sqrt( (x2 - x1)^2 + (y2 - y1)^2)


3. The attempt at a solution

Well I really don't think I'm doing this right. I tried substituting the given point and distance into the equation, but I'm not really sure where to go from there.

4*sqrt(17) = sqrt( (18 - x)^2 + (0 - y)^2)
272 = y^2 + x^2 - 36x + 324
y^2 = -(x-18)^2 + 272

I'm not really sure what to do. Any help would be greatly appreciated.

tiny-tim

hi theintarnets! welcome to pf!

that's ok so far …

now put y² = x⁴…

that gives you a quartic equation …

but you should be able to spot one root by guesswork (you only need one )

theintarnets

I'm sorry, I'm not very good at math and I have no idea what you mean.
So I substitute x^4 for y^2? Can you explain why? And I'm not 100% sure what you mean by finding the roots. I'm so ashamed

theintarnets

Nevermind, figured it out from here :D http://answers.yahoo.com/question/in...6145630AA2WRyJ

tiny-tim

hi theintarnets!

(just got up …)
hmm … that's a different problem, finding a point at the shortest distance rather than at an actual distance
this is a standard method for solving a pair of simultaneous equations (ie two equations with two unknowns, x and y) …

you rewrite one equation to express y in terms of x (or vice versa),

then you put that value of y into the other equation, giving you an equation with only one unknown (x)!
the solutions of a polynomial equation are also known as the roots …

in this case, there's an obvious whole number that works … try it!