Rectangular Coordinates

1. Distance Formula

The distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

2. Midpoint Formula

The midpoint of the line segment between the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $M=((x_1+x_2)/2,(y_1+y_2)/2)$.

3. Standard Equation of a Circle

The circle with center $(h,k)$ and radius $r$ has the equation $(x-h)^2+(y-k)^2=r^2$.

Ex 1 Find the center and radius of the circle with standard equation $x^2+6x+y^2-8y-11=0$.

Sol Completing the square on the x terms and the y terms gives $(x^2+6x+9)+(y^2-8y+16)=11+9+16$ or $(x+3)^2+(y-4)^2=36$, so the center of the circle is the point $(-3,4)$, and its radius is $6$.

Ex 2 Use the distance formula to find an equation of the perpendicular bisector of the line segment between the points $(4,3)$ and $(-2,5)$.

Sol The point $(x,y)$ is on the perpendicular bisector iff it is equidistant from the two points, so the perpendicular bisector is defined by the equation $\sqrt{(x-4)^2+(y-3)^2} = \sqrt{(x+2)^2+(y-5)^2}$. Squaring both sides gives $(x-4)^2+(y-3)^2 = (x+2)^2+(y-5)^2$, and then multiplying out both sides yields $x^2-8x+16+y^2-6y+9=x^2+4x+4+y^2-10y+25$; so the perpendicular bisector has equation $-12x+4y=4$ or $y=3x+1$.

Pr A Find an equation of the circle with center at the origin which passes through the point $P(5,-3)$.

Pr B Find an equation of the circle which has the midpoint of the line segment from $A(4,-1)$ to $B(2,5)$ as its center and has radius $r=9$.

Pr 1 Find the distance from the point $P(-3,5)$ to the midpoint of the line segment between $A(5,-2)$ and $B(1,6)$.

Pr 2 Find an equation of the circle with center $C(7,-5)$ which is tangent to the x-axis.

Pr 3 Find an equation of the circle with center $C(3,5)$ which passes through the point $P(1,2)$.

Pr 4 Find an equation of the circle with center $C(-2,4)$ which is tangent to the line $x=3$.

Pr 5 Find an equation of the circle which has the line segment from $P(-2,3)$ to $Q(6,-1)$ as a diameter.

Pr 6 Use the distance formula to determine if the point $P(8,2)$ is inside, outside, or on the circle with equation $x^2-6x+y^2+8y-39=0$.

Pr 7 Find the point on the circle with equation $(x+3)^2+(y-2)^2=20$ which is closest to the point $P(1,-6)$.

Pr 8 Calculate the distance from the point $P(1,3)$ to the line $3x-4y=21$.

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