(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

There are two parabolas which intersect.

One is y = x^2

The other is the same size, but is on a rotated cartesian plane, 45º CW of the first parabola.

Find the the highest intersection point.

2. Relevant equations

Anything relevent.

3. The attempt at a solution

First of all, I see that the fact there is an angle being used. So I think we'd need to get the parabolas in another co-ordinate system based on angles. Polar co-ordinates are there for us.

So I construct a triangle with x, y, r and theta. I need to get r in terms of theta.

For one, tan0 = y/x

y = x^2, as it's a parabola.

So

[tex]

tan\theta = x^2 / x,

tan\theta = x,

r = \sqrt{x^2 + y^2},

r = \sqrt{x^2 + (x^2)^2},

r = \sqrt{x^2 + x^4},

r = \sqrt{tan^2\theta + tan^4\theta}[/tex]

So by changing the theta, I could rotate the parabola.

By adding 45º, (pi / 4), it would rotate it to the correct position.

So I need to solve:

[tex]

\sqrt{tan^2\theta + tan^4\theta} = \sqrt(tan^2(\theta + \pi/4) + tan^4(\theta + \pi/4))[/tex]

Here I get stuck. I can't seem to solve this.

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# Homework Help: Parabolas that intersect, tricky one

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