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I know I am supposed to ask a specific question, but there's just too many that I would have to ask [it would be like spam] since I missed one week of school because of a family thing and we have an exam this Tuesday, and the teacher's got Mondays off, since it's only 'practice' classes, which means I can't ask her. So, I will just group them here, hoping someone will answer.


Lines

Explicit, Implicit and Segment Line equation

Let's say you have this line equation (implicit form): $x-2y-3=0$

How to convert that (back and forth) into explicit and segment forms.

Common point / Line crossing point

You have two lines: $x-2y-3=0$ and $3x-2y-1=0$

How do you determine where they cross (and if they cross) [This might be a bad example].

Angles between lines

So, taking the two lines from the above example: $x-2y-3=0$ and $3x-2y-1=0$

How would you determine the angle between them (if they're not parallel, that is).

$k$ - the direction coefficient

When given the following line equation: $3x-2y-1=0$. How does one calculate $k$?


Circles

Writing the 'proper' circle equation

I know, the title is a bit... odd, but I will provide an example.

Let's say you're given this circle equation: $x^2+y^2+6x-2y=0$

That has to be transformed into something that resembles: $(x-p)^2+(y-q)^2=r^2$

I would take this approach: $x^2+y^2+6x-2y=0$ / $+3^2-3^2+1-1$

When sorted out you get: $x^2+6x+3^2+y^2-2y+1=8$ which is in fact: $(x+3)^2+(y-1)^2=8$. I hope I'm right! :P

Defining whether a point is a part of the circle

Let's have we have this circle: $(x+3)^2+(y-1)^2=8$, how would you define whether point $T$ is a part of the circle's 'ring.' I'm going on a limb here, and I'll just point out a thought: Would you just replace the $x$ and $y$ in the equation with the coordinates of $T$?

Tangent

This one's a little tougher (at least I think so). So, you have $(x+3)^2+(y-1)^2=8$ and a point $T(-2,4)$ which can be on or off the circle. Now, I know there's 2 approaches: One if the point is on the circle and the second one if it's off it. So, you have to write a Line equation of the Tangent that goes through that point. I really couldn't figure this one at all I have a vague idea of how to do all the above mentioned, but this one's a bit a total mess.

Circle equation of a circle that touches both of the axis and the circle's centre point lies on a given line

Whew, that took a while to compose... Well, Let's say we have the line $x-2y-6=0$ and we have to determine the centre and the equation of the circle, taking into consideration that the circle touches both axis. The only thing I can gather from that is that $|q|=|p|=r$


Well, I hope someone actually reads and answers this, because I've been writing it for the past hour flipping through the textbook like a madman. And it would save my life.

Thanks!

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    Ok: so these feel lot classwork, and it would be best if you showed some of your own work. This is for two reasons: it shows us that you aren't using us just for answers, and it shows us the methods that you are allowed to use. A quick question: do you know dot products or anything of vectors? And if these are homework, please say so.2011-06-05
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    Now, 2 clarifications: [1] What do you mean by explicit, implicit, and segmented forms? [2] Is a k-direction coefficient the same thing as 'slope?'2011-06-05
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    @mixedmath Well, we did vectors a while back, but I don't think we're allowed to use them. If it were homework I would've told you so :.). I can show you some examples of what they did while I was gone. Well, actually the "writing proper circle equation" was a part of this weeks homework. I'll edit question and let you know in the comments when I'm done.2011-06-05
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    Observe that $x-2y-3=0\Leftrightarrow y=\frac{1}{2}x-\frac{3}{2}$.2011-06-05

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