Answer:
The standard form of the equation of the line is √3 x + y = 0
Explanation:
* Lets explain the problem
- The length of the line (r) which passes through the points (0 , 0) and
(x , y) is r = √(x² + y²)
- The direction of the line with the positive part of x-axis is Ф
- There is a line passing through the origin (0 , 0) and the point (x , y)
∵ r = √(x² + y²)
∵ r = 8
∴ √(x² + y²) = 8 ⇒ square the two sides
∴ x² + y ² = 64
∵ cos Ф = x/r and sin Ф = y/r
∴ x = r cos Ф and y = r sin Ф
∵ r = 8 and Ф = 120°
∵ cos 120° = -1/2
∵ sin 120° = √3/2
∵ x = r cos Ф
∴ x = 8 (-1/2) = -4
∵ y = r sin Ф
∴ y = 8 (√3/2) = 4√3
∴ The line passes through the origin (0 , 0) and point (-4 , 4√3)
- The equation of the line is y = mx + c, where m is the slope of the line
and c is the y-intercept (the line intersects y-axis at (0 , c))
- The slope of a line which passes through the origin and the
point (x , y) is m = y/x
∴ m = 4√3/-4 = -√3
∴ The equation is y = -√3 x+ c
- The line passes through the origin (0 , 0)
∴ c = 0
∴ The equation is y = -√3 x
- The standard form of the linear equation is Ax +By = C, where A , B , C
are constant
∵ y = -√3 x ⇒ add both sides by -√3 x
∴ √3 x + y = 0
* The standard form of the equation of the line is √3 x + y = 0