Express by an algebraic equation the statement that the line joining P(x,y)to the point (12,-5) has an inclination of arctan 1/3.
please help...thanks
Did you know that the slope of a line is the tangent of the angle that the line makes with the x-axis?
So they are telling us that the slope of the line is 1/3
so (y+5)/(x-12) = 1/3
cross-multiply and simplify if you want the actual equation of the line.
To express the statement that the line joining point P(x, y) to the point (12, -5) has an inclination of arctan(1/3), you can use the slope-intercept form of a line.
The slope-intercept form of a line is given by:
y = mx + b
where m represents the slope of the line and b represents the y-intercept.
First, let's find the slope (m) of the line. The slope of a line is equal to the tangent of the angle of inclination. In this case, the angle of inclination is arctan(1/3).
Since the tangent (tan) of an angle is equal to the ratio of the opposite side to the adjacent side, we can determine the slope as:
m = tan(arctan(1/3))
Simplifying, we have:
m = 1/3
Now we have the slope (m) of the line. To find the equation of the line, we need to find the y-intercept (b). We can substitute the coordinates of one of the points on the line (in this case, P(x, y)) and the slope (m) into the slope-intercept form.
Using the point (x, y), we have:
y = mx + b
y = (1/3)x + b
Now, let's substitute the coordinates of the point P(x, y) = (x, y) into the equation:
y = (1/3)x + b
y = (1/3)x + b
Since we know that P(x, y) lies on the line, we can substitute the specific coordinates of the point (12, -5) into the equation:
-5 = (1/3) * 12 + b
Now, solve for b:
-5 = 4 + b
b = -9
Therefore, the equation of the line joining point P(x, y) to the point (12, -5) with an inclination of arctan(1/3) is:
y = (1/3)x - 9