Finding Two Points on a Graph with Midpoint (0,0)

[Denyven]

Homework Statement

Two points are located on the graph $$y=4x^{2}+7x-1$$. A line drawn between these two points have a mid-point at (0,0). Find these two points.

Homework Equations

The midpoint formula $$(x_{m},y_{m})=(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$$

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 $$d=\sqrt{16x^{4}+56x^{3}+42x^{2}-14x+1}$$, by plugging in the parabolic equation into the $$d=\sqrt{x^{2}+y^{2}}$$. I have also figured out these set of rules for $$x_{1}, x_{2}, y_{1}$$ and $$y_{2}$$:
$$x_{1}+x_{2}=0$$ and $$y_{1}+y_{2}=0$$
Thus $$x_{1}= -x_{2}$$ and $$y_{1}= -y_{2}$$
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 don't 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 $$x_{1}+x_{2}=y_{1}+y_{2}$$
Or plug the equation of a parabola into the y1+y2?
Which would yield $$y=8x^{2}+14x-2$$, who's zeros are $$x=\frac{1}{8}(-7-\sqrt{65})$$ and $$x=\frac{1}{8}(\sqrt{65}-7)$$.
Are these the x values of either points?

 

[tiny-tim]

Tiny Tim,
What do you mean convert y1+y2 into x1+x2?

No, I said x1 and x2.

… Or plug the equation of a parabola into the y1+y2?
Which would yield $$y=8x^{2}+14x-2$$, who's zeros are $$x=\frac{1}{8}(-7-\sqrt{65})$$ and $$x=\frac{1}{8}(\sqrt{65}-7)$$.
Are these the x values of either points?

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