4
Find the equation of the perpendicular bisector of
the line joining the points (4,8) and (8,16)
s. Find the equation of the line passing through
(2,1) and perpendicular to the line -3x= 4y + 7
Find the equation of the line passing through
(3, 2) and perpendicular to the line joining the
points (s, 1) and (1,0).
Find the equation of the line passing through
(-3, -2) and parallel to 4y = 5x-9.

a) x+2y-30 = 0

b) 3y = 4x-5

c) y+4x = 14

d) 4y = 5x + 7

Step-by-step explanation:

First find the gradient of the line joining the two points (4,8) and (8,16):

m1 = y2 - y1/x2 - x1 = 16 - 8/8 - 4 = 8/4 = 2

From the law of perpendicularity, m1m2 = -1;

Gradient of the unknown line, m2 = -1/2

Also, the midpoint of a line = (x2+x1/2, y2+y1/2)

Therefore midpoint of the given line = (4+8/2, 8+16/2)

= (12/2, 24/2) = (6,12)

Hence, equation of the perpendicular bisector is

y-y1 = m(x-x1) where m = m2 and (X1,X2) = (6,12)

y-12 = -1/2(x - 6)

y - 12 =-1x/2 + 3

y = -x/2 + 3+12

y = -x/2 + 15 (Q.E.D)

which can be in different forms as follows

Multiply through by 2

2y = -x+30

2y+x=30

x+2y = 30

x+2y-30 = 0

b) From the equation, -3x = 4y+7

-4y=3x+7

divide through by -4

y=-3x/4 - 7/4

By comparing with y = mx + c, we have

m1 = -3/4 and from perpendicularity rule, m1m2 = -1

Therefore, gradient of the unknown line, m2 = 4/3

Hence, The equation is

y-y1 = m2(x-x1)

y - 1 = 4/3(x-2)

open the bracket by multiply its elements by 4/3

y-1 = 4x/3 - 8/3

Multiply through by 3

3y-3 = 4x -8

3y=4x-8+3

3y=4x-5

c) m1 = y2-y1/x2-x1 = 0-1/1-5 = -1/-4 = 1/4

m2 = -1/m1 = -4

Equation is

y-y1 = m2 (x-x2)

y-2= -4(x-3)

y-2 = -4x+12

y=-4x+12+2

y=-4x +14

y+4x=14

d)From 4y = 5x-9

y = 5x/4 - 9/4

by comparing with y = mx +c

m1 = 5/4

For parallelism, m1 = m2

m2 = 5/4

y - y1 = m2(x-x1)

y-(-2) = 5/4(x-(-3))

y+2 = 5/4(x+3)

y+2 = 5x/4 + 15/4

Multiply through by 4

4y+8 = 5x + 15

4y = 5x +15-8

4y = 5x + 7

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step-by-step explanation: