Urgent Math Help!

Hi Katie,

For #1, the given points are known to not be the center of the circle. The problem says "A circle in the xy-plane contains the points..." A circle can't contain it's center, the center is in the center of the circle, not on the perimeter of the circle which is what the equation describers. An equation can be written based of the center of a circle and its radius, but plugging in the center point into the circle's equation would result in an untrue equation.

#2, Subtracting equations is actually a common practice in solving systems of equations. In this problem we have a system of equations in three variables. The system can either be solved by using substitution or by adding/subtracting the equations to eliminate the variables. In this case the solution shows how to do this using the addition/subtraction method, and for good reason! Solving this system using substitution wouldn't be fun...

Hope this helps! Let me know if you have any further questions!

Tough question right there! My methods will probably take more than a minute to solve this problem. 

Since it is necessary to have an origin ( the centre of the circle) to write a circle equation, I will try to find it first. So the circle goes through 3 points, which means that this a circumscribed circle of this triangle. This would also mean that the intersect of the midperpendiculars of two sides of this triangle is the origin of the circle. Simple as that. However, the solution that I've just mentioned maybe a bit difficult because it involves types of Vector which I'm not sure whether they're taught to high school students in your country. 

Don't worry if you haven't got that one. There is still one more approach that I can think of right now - this one (I think) is rather easier and even faster. So let's call the center of the circle: O (a,b), then you let the distance of O to one point eg: (1,1) equal the distance of O to another point eg: (-1,1). You'll get a=0 by now. Do the same thing again, let the distance of O to the point (1,1) equal the distance of O t (-1,-1), replace a with 0 this time and you'll get b which is also zero. When you got a and b (coordination of O), you may as well plug them back into the distance equation that you've written before, and this time you get the radius.  

Since not all countries in this world have the same subject curriculum, I really hope my strange methods are useful to you ( In fact, they're widely used by me and my friends in Vietnam when it comes to these kinds of problems ). So that's all, if you have any questions, feel free to ask me. I'll try my best to answer them.