Line L1passes through the points A(1,2) and B(3,-4).Find the equation of L2 passing through the mid-point of AB and perpendicular to L1,leaving your answer in the form ax+by+c=0
L1 has slope 3
so L2 has slope -1/3
The midpoint is at (2,-1)
So, now we have a point and a slope, making the equation
y+1 = -1/3 (x-2)
To find the equation of line L2 passing through the mid-point of AB and perpendicular to line L1, we can follow these steps:
Step 1: Find the midpoint of AB.
To find the midpoint, we use the formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Given A(1, 2) and B(3, -4), we substitute the values into the formula:
Midpoint = ((1 + 3)/2, (2 + (-4))/2)
= (4/2, -2/2)
= (2, -1)
So, the midpoint of AB is (2, -1).
Step 2: Determine the slope of line L1.
The slope of line L1 can be found using the formula:
Slope (m1) = (y2 - y1) / (x2 - x1)
Given A(1, 2) and B(3, -4), we substitute the values into the formula:
Slope (m1) = (-4 - 2) / (3 - 1)
= -6 / 2
= -3
So, the slope of line L1 is -3.
Step 3: Find the slope of line L2 (since L2 is perpendicular to L1).
To find the slope of L2, we use the fact that the product of the slopes of perpendicular lines is -1.
So, the slope of L2 (m2) can be calculated as:
m2 = -1 / m1
Given the slope of L1 as -3, we substitute the value into the formula:
m2 = -1 / (-3)
= 1/3
So, the slope of line L2 is 1/3.
Step 4: Find the equation of line L2 using the slope-intercept form.
The equation of line L2 can be expressed in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Since L2 passes through the midpoint (2, -1), we substitute this point and the slope (1/3) into the slope-intercept form:
-1 = (1/3)(2) + b
Simplifying the equation:
-1 = 2/3 + b
To isolate b, we subtract 2/3 from both sides:
-1 - 2/3 = b
-3/3 - 2/3 = b
-5/3 = b
So, the y-intercept (b) of line L2 is -5/3.
Step 5: Convert the equation of L2 to the standard form ax + by + c = 0.
To convert the equation to the standard form, we move all the terms to the left-hand side and multiply by the common denominator to eliminate fractions.
y = mx + b
Multiply by 3 to eliminate the fraction:
3y = (1/3)x + (-5/3)
3y - (1/3)x = -5/3
To convert the equation to the standard form ax + by + c = 0, we multiply through by 3:
9y - x = -5
So, the equation of line L2 in the standard form ax + by + c = 0 is 9y - x = -5.