Expert: Steve Holleran - 9/20/2007

Question
Below is what my book said:

In this project you will find the points of intersection of the circle and porabola given by...

x^2+y^2-3x+5y-11=0   and  y=x^2-4x+5

a) Begin by writing the circle as the union of two functions.  Identify the functions that represent the top hald and the bottom half of the circle.



Can you PLEASE help me!!!!

due tomorrow ... i have been trying to figure this out all night!!

Answer
Hi Tay,

Well, I'll do what I can here, but I'm not sure how much help it'll be.  This is an amazingly ugly problem.

On the circle, complete the square in x and y:

 x^2 - 3x      + y^2 - 5y      = 11

 (x^2 - 3x + 9/4) + (y^2 -5y + 25/4) = 11 + 9/4 + 25/4

 (x - 3/2)^2 + (y - 5/2)^2 = 78/4

So,        (y - 5/2)^2 = 78/4 - (x-3/2)^2

Taking square roots,

            y - 5/2 = +/- sqrt[78/4 - (x-3/2^2]

   and      y = 5/2 +/- sqrt[78/4 - (x-3/2)^2]

Now the only thing I can think that they mean by writing this as the union of two functions would be to separate it at the+/- sign:

Upper half:   y = 5/2 + sqrt[78/4 - (x-3/2)^2]

Lower half:   y = 5/2 - sqrt(78/4 - (x-3/2)^2]

Like I said, this is really ugly!!!

I tried to find the intersection points algebraically, but you get a 4th degree polynomial that is not solveable over the rationals, so the only way I could find the points was on a graphing calculator (TI-83+).

I got :   (2.841, 1.707)    and (1.055, 1.893)

Maybe that's where the project is headed, because to try to do it algebraically, you would substitute the parabola equation into the circle equation for y, and this gets really bad.


I hope this is what you needed Tay, and I hope I got it to you in time.

Steve Holleran