Geometry

Solutions

Problem Set #4

a)      center at (5,7) and radius of 7    (x-5)2  + (y-7)2  = 49

b)      center at (-2,3)  and radius of 9   (x+2)2  + (y-3)2  = 81

 

2.      What is the center and radius of the circle described by each equation below?

a.    x2 -8x +16 + y2 =25    ->  (x + 4)2 + y2 = 52  so center at (-4,0)  and radius = 5

b.      x2 + 2x + 1 + y2 – 6y + 9 = 49   ->  (x+1)2 + (y-3)2 = 72   so  (-1,3)  and R = 7

3.      What is the center and radius of the circle described by each equation below?

a)      x2 -10x + y2 +4y = 7      x2 -10x + 25 + y2 +4y +4  = 7   + 25  + 4  ->  (x-5)2 + (y+2)2 =36  

                                                so  (5,-2)  is the center and radius of 6

b)      x2  + y2 -8y = 33   x2  + y2 -8y + 16  = 33 + 16  -> x2  + (y -4)2 = 72   (0,4)  R=7

c)      x2  -18x + y2 -2y  + 81 = 0     x2  -18x  + 81 + y2 -2y  +1  = 0  + 1  ->  (x-9)2 + (y-1)2 = 1
                                                                                              the center is  (9,1) and R = 1

d)      x2  + 10x + y2 -14y  =  7   x2  + 10x+ 25  + y2 -14y + 49   =  7 + 25 + 49  ->  (x+5)2 +( y-7)2  = 81

                                                                                               the center is (-5,7)  and radius is 9

 

                        Comment:  in all 4 parts of Problem #3  the algebraic technique of “Completing the Square”  was used.  In a nutshell, to

complete the  square of an expression like

 

                                                            x2  +  ax           you take a/2, (half the x term coefficient)  square it and add it to both sides of the equation

 

                        Thus you have             x2  + ax  + a2/4     which is  (x + a/2)2.

 

                   Always remember that whatever you add to one side of the equation you must add to the other side or you no longer have an equation!