parallel and perpendicular lines
Determine an equation of each of the following lines .
*the line perpendicular to y-4=0 and passing through the point (-1,6)
** the line perpendicular to 3x-12y+16=0 and having the same y-intercept as the line 14x-13y-52=0
and explain ....
To find the equation of a line perpendicular to a given line, we need to consider the slope of the given line.
For the first question:
The given line, y - 4 = 0, is a horizontal line passing through y = 4. Since it is a horizontal line, the slope of this line is 0.
A line perpendicular to a horizontal line will be a vertical line. Therefore, the slope of the perpendicular line is undefined.
Now, we have the equation of the line passing through the point (-1,6) with an undefined slope, which means it is a vertical line. A vertical line passing through (-1,6) has the equation x = -1.
So, the equation of the line perpendicular to y - 4 = 0 and passing through (-1,6) is x = -1.
For the second question:
We have two given lines:
1. 3x - 12y + 16 = 0
2. 14x - 13y - 52 = 0
To determine the slope of the first line, let's rearrange it into slope-intercept form (y = mx + b):
3x - 12y + 16 = 0
-12y = -3x - 16
y = (1/4)x + (4/3)
From the equation, we can see that the slope of the first line is 1/4.
To determine the y-intercept of the second line (line 2), we can rearrange its equation into slope-intercept form:
14x - 13y - 52 = 0
-13y = -14x + 52
y = (14/13)x - (52/13)
From the equation, we can see that the y-intercept of line 2 is -52/13.
Now, we need to find a line perpendicular to line 1 with the same y-intercept as line 2.
Since line 1 has a slope of 1/4, the line perpendicular to it will have a slope that is the negative reciprocal, which is -4.
Using the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we can substitute the slope (-4) and the y-intercept (-52/13) to find the equation of the perpendicular line:
y = -4x - 52/13
Therefore, the equation of the line perpendicular to 3x - 12y + 16 = 0 and having the same y-intercept as 14x - 13y - 52 = 0 is y = -4x - 52/13.
To determine the equations of lines that are perpendicular or parallel to a given line, we need to understand the properties of these lines.
1. Perpendicular lines:
- Two lines are perpendicular if the product of their slopes is equal to -1.
- The slope of a line in the form y = mx + b is given by m, where m is the coefficient of x.
Now, let's solve each problem step by step:
Problem 1:
We have the line equation y - 4 = 0, which is in the form y = 4.
To find the slope of this line, we can rewrite it in slope-intercept form (y = mx + b), where m is the slope:
y = 0x + 4
The slope (m) of this line is 0.
Since the line we need to find is perpendicular, the slope of the new line will be the negative reciprocal of 0, which is undefined. This means the equation of the line will be x = a, where a is the x-coordinate of the given point (-1,6):
Therefore, the equation of the line perpendicular to y - 4 = 0 and passing through (-1,6) is x = -1.
Problem 2:
We have two lines to consider:
- 3x - 12y + 16 = 0, which can be rewritten as 3x - 12y = -16
- 14x - 13y - 52 = 0
Step 1: Find the slope-intercept form of each line.
- Line 1: 3x - 12y = -16
We can solve for y to get it in slope-intercept form:
-12y = -3x - 16
y = (1/4)x + (4/3)
The slope (m) of Line 1 is 1/4.
- Line 2: 14x - 13y - 52 = 0
We can rearrange the equation to isolate y:
-13y = -14x + 52
y = (14/13)x - 4
The slope (m) of Line 2 is 14/13.
Step 2: Find the y-intercept of Line 2, as it should be the same as the line perpendicular to it.
We observe that the y-intercept of Line 2 is -4.
Step 3: Find the slope of the line perpendicular to Line 2.
The slope of a line perpendicular to Line 2 is the negative reciprocal of the slope of Line 2.
The slope (m) of the line perpendicular to Line 2 is -13/14.
Step 4: Using the slope-intercept form (y = mx + b), we can write the equation of the line perpendicular to Line 2 with the same y-intercept.
y = (-13/14)x - 4
Therefore, the equation of the line perpendicular to 3x - 12y + 16 = 0 and having the same y-intercept as 14x - 13y - 52 = 0 is y = (-13/14)x - 4.
Put y-4 = o into the form
y = m x + b
y = 0 x + 4
now the slope of the line perpendicular to that is -1/m = -1/0 which is undefined so we will have to use our heads.
y = 4 is a horizontal line through y = 4, x = anything.
x = constant, any constant, is perpendicular to y = 4
So we want the vertical line x = -1
That line, x = -1 , passes through all values of y including 6
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Now
3x -12 y + 16 = 0
12 y = 3 x + 16
y = (1/4)x + 4/3
what is slope of perpendicular?
m' = -1/(1/4) = -4
so the line we want is
y = -4 x + b
what is b?
well when x = 0, -13 y = 52
y = -52/13
so
-52/13 = -4(0) + b
b = -52/13
so
y = -4x -52/13