What is the standard form of the equation of a line for which the length of the normal segment to the origin is 8 and the normal makes an angle of 120degrees with the positive x axis

The standard form of the equation of the line is √3 x + y = 0

Step-by-step 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

rx and sx

step-by-step explanation: