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#

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 necessary line equation 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
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Answer 2

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

Let's assume that the circle's standard equation has radius r and a center at origin.

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

The chord AB's coordinates for its end points

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

The middle point C (x',y') coordinate for 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 AB slope

#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)#

The perpendicular bisector of AB has a slope of

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

The formula for AB's perpendicular bisector

#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)#

Since this is, of course, the equation of a straight line through the circle's origin (0,0), or center, the perpendicular bisector of the chord equals the circle's diameter.

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Answer 3

The equation of the perpendicular bisector of a chord of a circle can be found using the midpoint formula and the negative reciprocal of the slope of the chord. Given the endpoints of the chord (x1, y1) and (x2, y2), the midpoint (h, k) of the chord is found using: h = (x1 + x2) / 2 k = (y1 + y2) / 2

Then, the slope (m) of the chord is found using: m = (y2 - y1) / (x2 - x1)

The negative reciprocal of the slope is: m_perpendicular = -1 / m

Using the point-slope form of a line (y - k = m_perpendicular * (x - h)), substituting the midpoint (h, k) and the negative reciprocal slope (m_perpendicular), we get the equation of the perpendicular bisector of the chord.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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