Q:

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

Accepted Solution

A:
Answer:The standard form of the equation of the line is √3 x + y = 0Step-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