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.
m = tanA = (-5 - Y) / (12 - X) = 1/3,
The slope of a line is equal to the
tangent of the angle between the line
and positive X-axis.
tanA = 1/3,
A = 18.4 Degrees.
To express the statement that the line joining the point P(x, y) to the point (12, -5) has an inclination of arctan(1/3) in an algebraic equation, we need to find the equation of the line in point-slope form.
The inclination of a line is the angle it makes with the positive x-axis. The tangent of the angle of inclination (θ) is equal to the slope of the line. In this case, the slope is given by the tangent of arctan(1/3), which is 1/3.
Using the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute (12, -5) for (x1, y1) and 1/3 for m:
y - (-5) = (1/3)(x - 12)
Simplifying the equation gives:
y + 5 = (1/3)x - 4
Finally, rearranging the equation to isolate y:
y = (1/3)x - 4 - 5
y = (1/3)x - 9
Therefore, the algebraic equation for the line joining P(x, y) to the point (12, -5) with an inclination of arctan(1/3) is y = (1/3)x - 9.
To express the statement algebraically, we need to consider the slope of the line joining the point P(x, y) to the point (12, -5).
Let's first calculate the slope:
The slope of a line is given by the tangent of the angle it makes with the positive x-axis. In this case, the angle is the arctan(1/3).
So, we have:
slope = tan(arctan(1/3))
slope = 1/3
Now, using the point-slope form of the equation for a line, we can find the equation:
(y - y1) = m(x - x1)
where m is the slope and (x1, y1) is a point on the line. In our case, the point (x1, y1) is (12, -5).
So, the equation of the line is:
(y - (-5)) = (1/3)(x - 12)
Simplifying, we get:
y + 5 = (1/3)x - 4
Rearranging the equation:
y = (1/3)x - 9
Therefore, the algebraic equation representing the line joining P(x, y) to the point (12, -5) with an inclination of arctan(1/3) is:
y = (1/3)x - 9