Let (x1,y1) be a solution, other than (0,0), of a direct variation equation. Write an second direct variation equation whosegraph is perpindicular to the graph of the first equation.
To write a perpendicular direct variation equation to the given equation, we need to understand the concept of perpendicular lines.
Perpendicular lines have slopes that are negative reciprocals of each other. The slope of a direct variation equation is constant, so to find the slope, we can use the given solution (x1, y1).
The general form of a direct variation equation is y = kx, where k is the constant of variation.
Given the point (x1, y1), we can find the slope of the initial direct variation equation:
m1 = y1 / x1
To find a perpendicular line, we need to find the negative reciprocal of m1, which is the negative inverse of the slope:
m2 = -1 / m1
Now that we have the slope of the desired perpendicular line, we can write the equation in the form of a direct variation equation.
Let's call the new equation y2 = k2x.
Since we have the slope m2, which is -1 / m1, we can substitute the slope of the initial equation into the new equation:
y2 = (-1 / m1) * x
This is the equation of the perpendicular direct variation line.
Let's take an example to illustrate this:
Suppose the given direct variation equation is y = 2x.
If we choose a point on this line, for example, (2, 4), the slope (m1) would be:
m1 = y1 / x1 = 4 / 2 = 2
Since we want the perpendicular line, we find the negative reciprocal of m1:
m2 = -1 / 2 = -1/2
Therefore, the equation of the perpendicular direct variation line would be:
y2 = (-1/2) * x
This is your second direct variation equation whose graph is perpendicular to the graph of the given equation.