**Rectangular Coordinates**

1. __Distance Formula__

The distance between the points and is given by .

2. __Midpoint Formula__

The midpoint of the line segment between the points and is given by .

3. __Standard Equation of a Circle__

The circle with center and radius has the equation .

**Ex 1** Find the center and radius of the circle with standard equation
.

**Sol** Completing the square on the x terms and the y terms gives
or
, so the center of
the circle is the point , and its radius is .

**Ex 2** Use the distance formula to find an equation of the perpendicular
bisector of the line segment between the points and .

**Sol** The point is on the perpendicular bisector iff it is
equidistant from the two points, so the perpendicular bisector is defined by
the equation
. Squaring both
sides gives
, and then multiplying out both
sides yields
; so the perpendicular
bisector has equation or .

**Pr A** Find an equation of the circle with center at the origin which passes
through the point .

**Pr B** Find an equation of the circle which has the midpoint of the line
segment from to as its center and has radius .

**Pr 1** Find the distance from the point to the midpoint of the
line segment between and .

**Pr 2** Find an equation of the circle with center which is tangent
to the x-axis.

**Pr 3** Find an equation of the circle with center which passes
through the point .

**Pr 4** Find an equation of the circle with center which is tangent
to the line .

**Pr 5** Find an equation of the circle which has the line segment from
to as a diameter.

**Pr 6** Use the distance formula to determine if the point is
inside, outside, or on the circle with equation
.

**Pr 7** Find the point on the circle with equation
which
is closest to the point .

**Pr 8** Calculate the distance from the point to the line .

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