Question

What is the equation of the perpendicular bisector of a chord of a circle?

Answer 1

For a chord AB, with `A(x_A,y_A) and B(x_B,y_B)`. the equation of the perpendicular bisector is `y=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+y_B)/2)+(y_A+y_B)/2`

Explanation:

Supposing a chord AB with
`A(x_A,y_A)`
`B(x_B,y_B)`

The midpoint is
`M((x_A+x_B)/2,(y_A+y_B)/2)`

The slope of the segment defined by A and B (the chord) is
`k=(Delta y)/(Delta x)=(y_B-y_A)/(x_B-x_A)`
The slope of the line perpendicular to the segment AB is
`p=-1/k` => `p=-(x_B-x_A)/(y_B-y_A)`

The equation of the line required is

`y-y_M=p(x-x_M)`
`y-(y_A+y_B)/2=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)`
Or
`y=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)+(y_A+y_B)/2`

Note: the center of the circle, assummedly point C`(x_C,y_C)`, is in the line defined by the aforementioned equation, with radius `r=AC=BC`. Therefore the perpendicular bisector of the question is also the locus of the possible centers of circles with the chord AB

Answer 2

The equation of the perpendicular bisector of a chord of a circle is the equation of a diameter of the circle.

Explanation:

Let the equation of the circle be standard one having center at origin and radius r

`color(blue)("The equation of the circle: "x^2+y^2=r^2)`

The coordinates of the end points of the chord AB

`color(red)(A->(rcosa,rsina)" & " B->(rcosb,rsinb)`

The coordinate of the middle point C (x',y') of AB

`x'=r/2(cosa+cosb)=rcos((a+b)/2)cos((a-b)/2)`
`y'=r/2(sina+sinb)=rsin((a+b)/2)cos((a-b)/2)`

`:.(y')/(x')=tan((a+b)/2)`

The slope of AB

`m=(rsina-rsinb)/(rcosa-rcosb)`

`=-(2cos((a+b)/2)sin((a-b)/2))/(2sin((a+b)/2)sin((a-b)/2))`

`=-cot((a+b)/2)`

Slope of the perpendicular bisector of AB is

`m'=-1/m=tan((a+b)/2)=(y')/(x')`

`=>m'x'=y'`

The equation of the perpendicular bisector of AB

`y-y'=m'(x-x')`

`=>y=m'x-m'x'+y'`

`=>y=m'x-y'+y'`

`=>y=m'x`

`y=tan((a+b)/2)x`

OR

`color(green)(y=(y')/(x')x)`

Obviously this is the equation of a straight line passing through the origin (0,0),the center of the circle. So the perpendicular bisector of the chord is a diameter of the circle.