A and B are the point (3,-1) and (2,3) respectively. L is the line 8y-2x-3=0. Show that;

(a) l is perpendicular to AB.
(b) l passes through the mid point of AB

Solve step by step

To show that the line L is perpendicular to AB, we can use the concept of slopes.

(a) Perpendicular lines have slopes that are negative reciprocals of each other. So, we need to find the slopes of both line L and line AB and show that they multiply together to give -1.

The equation of line L is given as 8y - 2x - 3 = 0. To find the slope of the line L, we can rearrange the equation into the slope-intercept form y = mx + c, where m is the slope:

8y - 2x - 3 = 0
8y = 2x + 3
y = (2/8)x + 3/8
y = (1/4)x + 3/8

Comparing this equation with y = mx + c, we find that the slope of line L is m = 1/4.

Now, let's find the slope of line AB using the two given points, (3, -1) and (2, 3). The slope can be calculated using the formula: m = (y2 - y1) / (x2 - x1).

m = (3 - (-1)) / (2 - 3)
m = 4 / (-1)
m = -4

The slope of line AB is -4.

To prove that line L is perpendicular to line AB, we need to show that (-4) * (1/4) = -1.

(-4) * (1/4) = -1

Since (-4) * (1/4) = -1, this shows that the lines L and AB are perpendicular to each other.

(b) To show that line L passes through the midpoint of AB, we can calculate the midpoint of AB and check if its coordinates satisfy the equation of line L.

The midpoint of AB can be found using the formula: ((x1 + x2) / 2, (y1 + y2) / 2).

Midpoint = ((3 + 2) / 2, (-1 + 3) / 2)
Midpoint = (5/2, 2/2)
Midpoint = (5/2, 1)

Now, let's check if the coordinates of the midpoint satisfy the equation of line L:

8(y) - 2(x) - 3 = 0
8(1) - 2(5/2) - 3 = 0
8 - 5 - 3 = 0
0 = 0

Since the equation 0 = 0 holds true, this means that line L passes through the midpoin

given line: 8y = 2x+ 3

y = (2/8)x + 3/8
y = (1/4)x + 3/8

The slope of the given line is 1/4

a) find the slope of AB and show it is equal to -4

b) midpoint of AB = ( (3+2)/2 , (-1+3)/2 )
or ( 5/2 , 1)
--- show that ( 5/2 , 1) satisfies the given equation by subbing it in to see that the statement is true.