SOLUTION: a) Find the radii of the circles x^2 + y^2 = 2 and (x-3)^2 + (y-3)^2 = 32 b) Find the distance between the centers of the circles c) Explain why the circles must be internally ta

a) Find the radii of the circles x² + y² = 2 and (x-3)² + (y-3)² = 32

The circle with center (h,k) has equation (x-h)² + (y-k)² = r²

If we write the first circle's equation as

                                          (x-0)² + (y-0)² = 2
                                                        _
we see that it has center (h,k) = (0,0) and radius r = Ö2   

From the second circle's equation         (x-3)² + (y-3)² = 32
                                                    __    ____     _ 
we see that it has center (h,k) = (3,3) and radius Ö32 = Ö16·2 = 4Ö2
                    
b) Find the distance between the centers of the circles

We use the distance formula, which says that the distance between two
points (x1, y1) and (x2, y2) is

                               _______________________
                          d = Ö(x2 - x1)² + (y2 - y1)²                         

So the distance between (x1,y1) = (0,0) and (x2,y2) = (3,3) is
                               ___________________
                          d = Ö(3 - 0)² + (3 - 0)²
                               _______
                          d = Ö3² + 3² 
                               _____
                          d = Ö9 + 9
                               __
                          d = Ö18 
                               ___
                          d = Ö9·2     
                                _
                          d = 3Ö2

c) Explain why the circles must be internally tangent

Because the distance between their centers plus the radius of the small
circle equals the radius of the large circle, since
                       _    _     _ 
                     3Ö2 + Ö2 = 4Ö2                 



Edwin
AnlytcPhil@aol.com