Find the equation of the straight line passing through the point(3,-2) and making an angle of 60° with the line√3x+y-1=0
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since the line √3x+y-1=0 makes angle of -30° with the x-axis, the desired line is inclined at an angle of +30°. So, we have
y+2 = 1/√3 (x-3)
y = 1/√3 x - (2+√3)
or, as the other line is given is standard form,
x-√3y-(3+2√3) = 0
To find the equation of the straight line passing through the point (3, -2) and making an angle of 60° with the line √3x + y - 1 = 0, you can follow these steps:
1. Convert the given equation to the slope-intercept form (y = mx + b), where m represents the slope, and b represents the y-intercept.
√3x + y - 1 = 0
y = -√3x + 1
2. Determine the slope of the given line. Comparing the equation with the slope-intercept form, we find that the slope is -√3.
3. Calculate the slope of the line that makes a 60° angle with the given line. Since the angles between two lines are equal to the angle between their slopes, the slope of the line making a 60° angle is the negative reciprocal of the given slope. So, the slope is -(1/(-√3)) = 1/(√3).
4. Use the point-slope form (y - y₁ = m(x - x₁)) to find the equation of the line passing through the given point (3, -2) with the calculated slope.
Substituting the values:
y - (-2) = 1/(√3)(x - 3)
y + 2 = 1/(√3)(x - 3)
5. Simplify the equation to get it in the desired form.
Multiplying both sides by √3 to eliminate the fraction:
√3(y + 2) = x - 3
Distribute √3 to the terms inside the parentheses:
√3y + 2√3 = x - 3
Rearrange the equation to match the standard form Ax + By + C = 0:
x - √3y - 2√3 - 3 = 0
Combining constants:
x - √3y - (2√3 + 3) = 0
Finally, the equation of the line passing through the point (3, -2) and making a 60° angle with the line √3x + y - 1 = 0 is:
x - √3y - (2√3 + 3) = 0