Two points on a graph, the mid-point of a line made by these two points is an origin.

Denyven

1. The problem statement, all variables and given/known data
Two points are located on the graph [tex]y=4x^{2}+7x-1[/tex]. A line drawn between these two points have a mid-point at (0,0). Find these two points.


2. Relevant equations
The midpoint formula [tex](x_{m},y_{m})=(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})[/tex]


3. The attempt at a solution
I have worked out the distance from a point on the graph to the origin as a function of x [tex]d=\sqrt{16x^{4}+56x^{3}+42x^{2}-14x+1}[/tex], by plugging in the parabolic equation into the [tex]d=\sqrt{x^{2}+y^{2}}[/tex]. I have also figured out these set of rules for [tex]x_{1}, x_{2}, y_{1}[/tex] and [tex]y_{2}[/tex]:
[tex]x_{1}+x_{2}=0[/tex] and [tex]y_{1}+y_{2}=0[/tex]
Thus [Tex]x_{1}= -x_{2}[/tex] and [tex]y_{1}= -y_{2}[/Tex]
All of the above were derived from the midpoint formula, since the mid-point is (0,0), both the x's and the y's have to cancel out each other.

Thanks in Advance!

um0123

This is interesting, i dont think its possible to find the where these points are located without any information about where at least one of them is.

but i may just not be looking close enough for a solution.

tiny-tim

Hi Denyven!

ok, so y₁ + y₂ = 0.

Now convert that equation into x1 and x2.

What do you get?

Denyven

Tiny Tim,
What do you mean convert y1+y2 into x1+x2?
Like this [tex]x_{1}+x_{2}=y_{1}+y_{2}[/tex]
Or plug the equation of a parabola into the y1+y2?
Which would yield [tex]y=8x^{2}+14x-2[/tex], who's zeros are [tex]x=\frac{1}{8}(-7-\sqrt{65})[/tex] and [tex]x=\frac{1}{8}(\sqrt{65}-7)[/tex].
Are these the x values of either points?

tiny-tim

No, I said x1 and x2.
What on earth are you doing?

What happened to x1 and x2?

Put them back!!

Denyven

oh ha,
so do you mean x1=y1+y2-x2 and x2=y1+y2-x1?

tiny-tim

No, I mean y₁ = 4x₁² + 7x₁ - 1

and y₂ = 4x₂² + 7x₂ - 1