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4 years ago

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Mathematics

1 answer:

Ira Lisetskai [31]4 years ago

6 0

Answer:

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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