Question

Find the point of intersection of y Axis and perpendicular bisector of 2, - 3 and -4, 1

Answer 1

The perpendicular bisector of (2, -3) and (-4, 1) passes through the centre of the line joining the two points

The co-ordinates of the centre = { (2+-4)/2 , (-3+1)/2}
                                                 = (-1, -1)

The slope/gradient of the line joining (2, -3) and (-4, 1) = (1--3)/(-4-2) = 4/-6 = -2/3

When two lines are perpendicular, the product of their slopes = -1

⇒ -2/3 x The slope of the perpendicular bisector = -1
⇒ The slope of the perpendicular bisector = 3/2

The equation of the perpendicular bisector; it passes through (-1, -1) and has a slope of 3/2

Taking any general point on the line (x, y)
Slope = (y--1)/(x--1) = 3/2
(y+1)/(x+1) = 3/2
y+1 = 3/2(x+1)
y+1 = 3/2x + 3/2
y = 3/2x + 3/2 - 1
y = 3/2x +1/2

The equation is now in the form y = mx + c

c= the y-intercept = 1/2

∴  the point of intersection of y Axis and perpendicular bisector of (2, - 3) and (-4, 1) is (0, 1/2)

(0, 1/2) is the answer