find the equation of a straight line on which the perpendicular from the origin makes an angle of 30 degree with the x-axis and which forms a triangle of are 50/root3 with the axes
if the base has length x,
area = 1/2 * x * x√3 = √3/2 x^2 = 50/√3
x = 10
so, y = 10√3
so, the line is
x/10 + y/(10√3) = 1
oops. x = 10/√3
y = 10
x/(10√3) + y/10 = 1
To find the equation of the straight line, we'll use the fact that the perpendicular from the origin makes an angle of 30 degrees with the x-axis.
Let's break down the problem into steps:
Step 1: Finding the slope of the line
The slope of the line can be found using the angle it makes with the x-axis. Since the perpendicular from the origin makes an angle of 30 degrees with the x-axis, the angle between the line and the x-axis will be 90 degrees - 30 degrees = 60 degrees.
The tangent of the angle between the line and the x-axis will be equal to the slope. In this case, we have tan(60 degrees) = sqrt(3). Therefore, the slope of the line is sqrt(3).
Step 2: Finding the equation of the line
The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
Since the line passes through the origin (0, 0), the y-intercept b will be 0. Therefore, the equation of the line becomes y = sqrt(3)x.
Step 3: Calculating the area of the triangle
The area of the triangle formed between the line and the axes can be calculated using the formula:
Area = (1/2) * base * height
In this case, the base is the x-coordinate of the point where the line intersects the x-axis, and the height is the y-coordinate of the point where the line intersects the y-axis.
To find the x-intercept of the line (where it intersects the x-axis), we set y = 0 in the equation:
0 = sqrt(3)x
Solving for x, we get x = 0.
To find the y-intercept of the line (where it intersects the y-axis), we set x = 0 in the equation:
y = sqrt(3)(0)
Therefore, the y-intercept is 0.
The area of the triangle is given as 50/sqrt(3), so we have:
(1/2) * 0 * 0 = 50/sqrt(3)
Simplifying, we find that the area is 0, which contradicts the given information. It's possible that there may be an error in the problem statement or the given area.
In summary, the equation of the straight line on which the perpendicular from the origin makes an angle of 30 degrees with the x-axis is y = sqrt(3)x. However, the given area of 50/sqrt(3) with the axes is not possible based on the calculations.