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.
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 $k$?
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 $T$?
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.