   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 .   