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
