(x-a)^{2} + (y-b)^{2} = r.
^{2} |
(3) |

A equation of the form

*Ax ^{2}* +

**Example:** Is 2*x ^{2}* + 4

The equation is now in standard form, and we can see that the center is at (-1,-3) and the radius is . To graph the circle, plot the center and the endpoints of the horizontal and vertical diameters.

**Exercises**

- 1.
- Give the standard equation for each of the following circles. Graph each
circle.
- (a)
- Center at (5,-3) and radius 5;
- (b)
- Center at (0,0) and radius 1;
- (c)
- Center at (-3, -7) and passing through the origin;
- (d)
- Center at the intersection of 3
*x*+2*y*= 5 and 2*x*- 7*y*= -5, radius 3.

- 2.
- Determine which of the following are circles by trying to put the
equations into standard form. Graph each circle, and if the equation is not
a circle explain why.
- (a)
*x*+ 2^{2}*x*+*y*+ 4^{2}*y*= 0;- (b)
- 2
*x*+ 4^{2}*x*+ 2*y*- 4^{2}*y*= 0; - (c)
- 4
*x*+ 2^{2}*x*+*y*+ 4^{2}*y*= 0; - (d)
- 4
*x*+ 2^{2}*x*+ 4*y*+ 4^{2}*y*= 40; - (e)
*x*+ 2^{2}*x*+*y*+ 4^{2}*y*+ 10 = 0;

- 3.
- Illustrate graphically the solution to the inequality
*x*+ 4^{2}*x*+*y*+ 12^{2}*x*< 0.

A sphere is the set of all points in space at a given distance (radius) from a
given point (center). The standard equation of a sphere with center (*a*,*b*,*c*)
and radius *r* is

(x-a)^{2} + (y-b)^{2} + (z-c)^{2} = r
^{2} |
(4) |

**Exercises:**

- 1.
- Give the standard form for the equation for each of the following
spheres:
- (a)
- Center at (0,0,0) and radius 1;
- (b)
- Center at (1,-2,3) and radius 8;
- (c)
- Center at (2,3,4) and containing (5,-3,5).

- 2.
- By completing the square, find the center and radius of the sphere with
the given equation:
- (a)
*x*+ 2^{2}*x*+*y*- 2^{2}*y*+*z*- 4^{2}*z*= 2;- (b)
*x*+ 4^{2}*x*+*y*+ 6^{2}*y*+*z*+ 8^{2}*z*= 0- (c)
*x*+^{2}*y*+ 2^{2}*z*= -*z*.^{2}

- 3.
- Describe geometrically the solution to the inequality
*x*+^{2}*y*-2^{2}*y*+*z*-4^{2}*z*> 0.