The figure shows an existing pipeline passing through two points. Plant WW is in existence at the point (4, 14). A new plant MM, is to be located on the Y-axis at a point such that the pipeline that will be constructed to connect Plant MM to the existing line will be perpendicular to the existing line in its intersection with the X axis. Give the position of the Plant MM (co-ordinate)
To find the position of Plant MM on the Y-axis, we need to consider the information provided in the question.
From the given information, we know that Plant WW is located at the point (4, 14). We also know that the pipeline connecting Plant MM to the existing line must be perpendicular to the existing line at its intersection with the X-axis.
A line that is perpendicular to another line has a slope that is the negative reciprocal of the slope of the original line.
To find the slope of the existing line, we can calculate the difference in the Y-coordinates divided by the difference in the X-coordinates between two points on the line.
Let's take two points on the existing line. We already have one: (4, 14). Let's assume the other point is (x, y).
The slope of the existing line is (14 - y) / (4 - x).
The slope of the new line (perpendicular to the existing line) will be the negative reciprocal of this slope.
So, the slope of the new line is -1 / [(14 - y) / (4 - x)].
Given that the new line intersects the X-axis, the Y-coordinate of this intersection point will be 0.
So, we can set the equation for the new line equal to 0 to find the X-coordinate of the intersection point:
0 = -1 / [(14 - y) / (4 - x)] * (x - 4)
Simplifying this equation, we get:
0 = - (4 - x) / (14 - y) * (x - 4)
Cross multiplying, we have:
0 = - (x - 4)^2 / (14 - y)
Now, to find the position of Plant MM on the Y-axis, we assume that its X-coordinate is 0, since it lies on the Y-axis.
Substituting x = 0 into the equation, we get:
0 = - (0 - 4)^2 / (14 - y)
Simplifying, we have:
0 = - (16) / (14 - y)
This equation implies that the denominator must be equal to 0 since division by 0 is undefined.
So, we have 14 - y = 0.
Solving for y, we find:
y = 14
Therefore, the position of Plant MM on the Y-axis is (0, 14).
To find the position of Plant MM, we need to determine the coordinates of its location on the y-axis. Let's analyze the given information step by step.
1. The existing pipeline passes through two points.
2. Plant WW is located at the point (4, 14).
3. The new plant MM needs to be located on the y-axis.
4. The pipeline connecting Plant MM to the existing line should be perpendicular to the existing line at its intersection with the x-axis.
To determine the location of Plant MM, we need to understand that a line perpendicular to another line has a slope that is the negative reciprocal of the slope of the original line.
Let's consider the existing line passing through points (4, 14) and (x₁, y₁). The slope of this line can be calculated using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Substituting the values of (4, 14) and (x₁, y₁), we get:
slope = (y - 14) / (x - 4)
Since the new line is perpendicular, we know that the slopes are negative reciprocals of each other. Therefore, the slope of the new line is -(1/slope). Let's call this slope m.
m = -(1 / slope)
To find the coordinates of Plant MM, we need to determine the x-intercept (where the line intersects the x-axis) of the new line. Let's call this point (x₂, 0).
Using the point-slope form of a line, we can write:
y - y₁ = m(x - x₁)
Substituting the values of (x₁, y₁) and (x₂, 0):
0 - 14 = m(x₂ - 4)
Simplifying this equation:
-14 = m(x₂ - 4)
Substituting the value of m, we have:
-14 = -(1 / slope) (x₂ - 4)
Cross-multiplying:
14 = (x₂ - 4) / slope
Multiplying both sides by the slope:
14 * slope = x₂ - 4
Finally, rearranging the equation for x₂, we get:
x₂ = 14 * slope + 4
Substituting the slope (-(1 / slope)) and evaluating, we can find the x-coordinate of Plant MM. To find the y-coordinate, we know it will be 0 as it lies on the y-axis.
Therefore, the coordinates of Plant MM are given by (x₂, 0).