Understanding the Equation of a Straight Line with a Given Angle of Inclination

When tackling a problem that involves finding the equation of a straight line given specific conditions such as passing through a point and forming an angle with another line, one must use a combination of geometric principles and algebraic techniques. In this article, we will solve a specific example: finding the equation of a straight line that passes through the point (1, 2) and forms an angle of 60° with the line ( sqrt{3}x y - 2 0 ).

Step 1: Finding the Angle of Inclination of the Given Line

To begin, we need to determine the angle of inclination of the given line. The given line is expressed in the standard form ( sqrt{3}x y - 2 0 ). Converting this to the slope-intercept form ( y mx c ) where ( m ) is the slope, we get:

[ y -sqrt{3}x 2 ]

Thus, the slope ( m ) of the given line is ( -sqrt{3} ). The angle of inclination ( theta ) can be found using the formula ( tan theta m ).

[ tan theta -sqrt{3} ]

Here, ( theta 120^circ ) since ( tan 120^circ -sqrt{3} ).

Step 2: Determining Possible Angles of Inclination for the Required Line

The required line can have two possible angles of inclination:

The angle of inclination of the given line plus 60°, which is ( 120^circ 60^circ 180^circ ). The angle of inclination of the given line minus 60°, which is ( 120^circ - 60^circ 60^circ ).

For each of these angles, we need to find the slope of the required line using the formula for the tangent of the difference between two angles:

[ tan(theta_1 pm theta_2) frac{tan theta_1 pm tan theta_2}{1 mp tan theta_1 tan theta_2} ]

Step 3: Finding the Slopes of the Required Line

For the angle of inclination of 180°: ( tan 180^circ 0 ) For the angle of inclination of 60°: ( tan 60^circ sqrt{3} )

Thus, the slopes of the required line are ( 0 ) and ( sqrt{3} ) respectively.

Step 4: Using the Slope-Point Form

The slope-point form of the equation of a straight line is given by:

[ y - y_1 m(x - x_1) ]

where ( (x_1, y_1) ) is a point on the line and ( m ) is the slope.

Given the point (1, 2), we can substitute this into the equation for each slope:

For the slope ( 0 ): ( y - 2 0(x - 1) ) ( y 2 ) For the slope ( sqrt{3} ): ( y - 2 sqrt{3}(x - 1) ) ( y sqrt{3}x - sqrt{3} 2 )

The two possible equations of the required line are:

( y 2 ) ( y sqrt{3}x - sqrt{3} 2 )

Conclusion

By utilizing the principles of trigonometry and the slope-point form of a line, we have successfully determined the equations of the straight line that satisfies the given conditions. This method can be applied to a wide range of similar problems in geometry and algebra, providing a robust approach to solving complex equations involving angles and lines.