EQUATION OF PERPENDICULAR BISECTOR

Example :

Find the equation of perpendicular bisector of the line joining the points A(-4, 2) and B(6, -4) in general form.

Solution :

Let E be the mid point and CD be the perpendicular bisector of AB. 

Mid point of AB : 

= E((x₁ + x₂)/2, (y₁ + y₂)/2)

Substitute (x₁, y₁) = A(-4, 2) and (x₂, y₂) = B(6, -4).

= E((-4 + 6)/2, (2 - 4)/2)

= E(2/2, -2/2)

= E(1, -1)

Slope of AB : 

= (y₂ - y₁)/(x₂ - x₁)

Substitute (x₁, y₁) = A(-4, 2) and (x₂, y₂) = B(6, -4).

= (-4 - 2)/(6 + 4)

= -6/10

= -3/5

Slope of AD : 

= -1/slope of AB

= -1/(-3/5)

= 5/3

Equation of the perpendicular bisector CD :  

y = mx + b

Substitute m = 5/3.

y = (5/3)x + b ----(1)

Substitute (x, y) = E(1, -1). 

-1 = (5/3)(1) + b

-1 = 5/3 + b

-1 - 5/3 = b

-8/3 = b

(1)----> y = (5/3)x - 8/3

Multiply each side by 5.

3y = 5x - 8

-5x + 3y + 8 = 0

5x - 3y - 8 = 0

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