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What is the angle made by the perpendicular from the origin to the line x+ √3y = 2 with the positive direction of x axis?
- a)60°
- b)135°
- c)30°
- d)45°
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
What is the angle made by the perpendicular from the origin to the lin...
- The given equation is
- x +√3y = 2
- On dividing both sides by
- [(−1)2 + (√3)2]1/2 = √4 = 2
- = x/2 + √3y/2 = 2/2
- Therefore , the equation becomes
- = (1/2)x + (√3/2)y = 1
- = x cos 60° + y sin 60° = 1
- Normal form of any line is : x cosw + y sinw = p
- w = 60°
Most Upvoted Answer
What is the angle made by the perpendicular from the origin to the lin...
Understanding the Line Equation
The given line equation is:
- x + √3y = 2
We can rearrange this into the slope-intercept form (y = mx + b) to find the slope:
- √3y = 2 - x
- y = (-1/√3)x + 2/√3
Thus, the slope (m) of the line is -1/√3.
Finding the Perpendicular Slope
The slope of a line perpendicular to another line is the negative reciprocal of the original slope:
- Perpendicular slope = √3
Finding the Angle with the X-axis
The angle θ that a line makes with the positive direction of the x-axis can be calculated using the tangent of the angle:
- tan(θ) = slope = √3
To find the angle θ:
- θ = tan^(-1)(√3)
This gives us:
- θ = 60°
Conclusion
Therefore, the angle made by the perpendicular from the origin to the line x + √3y = 2 with the positive direction of the x-axis is:
- 60° (Option A)
The given line equation is:
- x + √3y = 2
We can rearrange this into the slope-intercept form (y = mx + b) to find the slope:
- √3y = 2 - x
- y = (-1/√3)x + 2/√3
Thus, the slope (m) of the line is -1/√3.
Finding the Perpendicular Slope
The slope of a line perpendicular to another line is the negative reciprocal of the original slope:
- Perpendicular slope = √3
Finding the Angle with the X-axis
The angle θ that a line makes with the positive direction of the x-axis can be calculated using the tangent of the angle:
- tan(θ) = slope = √3
To find the angle θ:
- θ = tan^(-1)(√3)
This gives us:
- θ = 60°
Conclusion
Therefore, the angle made by the perpendicular from the origin to the line x + √3y = 2 with the positive direction of the x-axis is:
- 60° (Option A)
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