Hyperbola and the Line in the Cartesian coords.

[frostysh]

Homework Statement

We have the hyperbola, the focal stuff of which is on the Abscissa axis. $$x^2 - 2y^2 = 4 $$, and we have a line $$3x - 4y = 2$$, and we need to understand if this two crazy stuff will intersect, or be tangent, or nothing like the previous one.

Homework Equations

I don't clearly understand what is means "Relevant Equations".

The Attempt at a Solution

Well, first time I have made a mistake with a signs. In general, solution is next - we trying to find the points that belongs to each crazy stuff simultaneously, it's means this points a roots of the two equations in the same time.

I have made another few attempts, and in the one attempt, I have tried to change the order of the equations, and try to find a roots of Hyperbola, and then put it to the Line geometric-algebraic model.

Despite, and at least, when I have typing a request to this forum, I have found the true solution that fits with a book' answer.

so the question why despite of my few mistakes some wrong solutions satisfying the one of the equations in the each case?!

Thanx.

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Ooooops. My lot of pardons, I have realized why some wrong solution is satisfying some particular equations, that because in the first attempt $$x \in (- \infty , + \infty)$$, same as $$y \in (- \infty , + \infty)$$, and I have made mistake only in the equation of the Hyperbola, so if we going to put any number in the equation of the Line, and then with no mistake find the another coord. variable - we will obtain the right answer.
And in the second attempt, despite it's attempt was in the first coord. angle, due to avoiding less than zero values inside of the square root, we can say the same as we said about the Line equation. :) Me happy ^^.

 

[Mark44]

Everything in your images could be done in text using LaTeX. We have a tutorial here -- https://www.physicsforums.com/help/latexhelp/. You must be somewhat familiar with it, since the first two equations are done using it.
Some helpers won't respond if all work is shown as an image, as some handwritten work is difficult to read due to bad lighting of photographed images, or illegible writing. Some images we've seen have been posted sideways or upside-down. A major problem with large amounts of work posted as an image is that we can't insert a comment into an image at the point where an error is located. Instead, we have to include extra information about the region we're talking about.

 

[frostysh]

Hey, thanks for the answer, and for' the advises. I just don't like LATEX, it's making me headaches ^:/. Well, in this particular case, I think my images is good enough, at least I can understand what is inscribed there :P. Sec, I will fix the last two LATEX stuff according to the guide.

 

[Buffu]

Hey, thanks for the answer, and for' the advises. I just don't like LATEX, it's making me headaches ^:/. Well, in this particular case, I think my images is good enough, at least I can understand what is inscribed there :P. Sec, I will fix the last two LATEX stuff according to the guide.

You literally include everything in the post which is a waste of everyone's time. You should skip the parts where you do basic algebra and rather focus on main lines.

 

[Buffu]

I don't get the question properly but I think using IVT would be sufficient to prove that those two intersect.

 

[frostysh]

Nay, I am disagree, actually during a posting this "basic algebra" stuff I have found ma' mistake, so it was a very useful, at least for frostysh :P.
What is "IVT"? Nevermind, I will use a service of the allmighty GOOGLE :).

 

[Mark44]

What is "IVT"?

Intermediate Value Theorem