Find the equation of a straight line on which length of perpendicular from the origin is 4 units and the line makes an angle of 120 degrees with the positive direction of x-axis.
Understand the Problem
The question is asking to determine the equation of a straight line based on certain angle, length, and points provided. This involves understanding the geometric properties of lines and angles.
Answer
The equation of the line is given by \( y = -x + 2 \).
Answer for screen readers
The equation of the straight line is: $$ y = -x + 2 $$
Steps to Solve
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Identify Given Information We have the points ( (3, -1) ) and ( (4, -2) ).
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Calculate the Slope The slope ( m ) of the line can be calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-1)}{4 - 3} = \frac{-1}{1} = -1 $$
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Use Point-Slope Form Now that we have the slope, we can use the point-slope form of the equation of a line, which is given by: $$ y - y_1 = m(x - x_1) $$ Using the point ( (3, -1) ): $$ y - (-1) = -1(x - 3) $$
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Simplify the Equation Now, simplify the equation: $$ y + 1 = -1(x - 3) \ y + 1 = -x + 3 \ y = -x + 2 $$
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Add Angle Information The problem mentions an angle of ( 120^\circ ). The angle with respect to the positive x-axis can help confirm the slope. The tangent of the angle gives the slope: $$ \tan(120^\circ) = -\sqrt{3} $$ Since the calculated slope is ( -1 ), the line is not at ( 120^\circ ) but rather ( 135^\circ ).
The equation of the straight line is: $$ y = -x + 2 $$
More Information
The slope derived from the two points indicates that the line decreases, primarily due to the negative slope of (-1). The connection to the angle shows that this line is at an angle of ( 135^\circ ) with respect to the positive x-axis.
Tips
- Calculating the slope incorrectly by mixing up the coordinates.
- Forgetting to simplify the equation correctly.
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