Machinery's Handbook, 31st Edition
46 EQUATION FORMS OF A LINE Perpendicular Lines: The two lines, l 1 and l 2 , are perpendicular if the product of their slopes is - 1, that is, if m 1 m 2 = - 1, since m 1 = –1/ m 2 . (The slopes are negative recipro- cals of one another.) Example 7: (a) Find the equation of the line that passes through the point (3, 4) and is parallel to line 2 x - 3 y = 16. (b) Find the equation of the line perpendicular to the given line and through the same point. Solution (a): Line 2 x - 3 y = 16 in slope-intercept form is y = 2 ⁄ 3 x - 16 ⁄ 3 , so the equation of a line passing through (3, 4) is y – 4 m x – 3 ( ) = . Parallel lines have equal slope. Thus, from the point-slope form, y – 4 = 2 ⁄ 3 ( x – 3) is par- allel to line 2 x - 3 y = - 6 and passes through point (3, 4). Solution (b): As illustrated in part ( a ), line 2 x - 3 y = - 6 has a slope of 2 ⁄ 3 . The prod- uct of the slopes of perpendicular lines is - 1, thus the slope m of a line passing through point (3, 4) and perpendicular to 2 x - 3 y = - 6 must satisfy the following: m –1 m 1 --- -–- 1- 3 2 –-- = = = 2 ⁄ 3 The equation of a line passing through point (3, 4) and perpendicular to the line 2 x - 3 y = 16 is y - 4 = - 3 ⁄ 2 ( x - 3), which rewritten is 3 x + 2 y = 17. Angle Between Two Lines: For two non-perpendicular lines with slopes m 1 and m 2 , the angle θ between the two lines is found by first applying trigonometric equation: θ tan m 1 m 2 – 1 m 1 m 2 + = ------------ The discussion of how the angle is determined by this relation is found in TRIGONOM- ETRY , which begins on page 94. Example 8: Find the angle between the lines: 2 x - y = 4 and 3 x + 4 y = 12. Solution: Rearranging each to be in the slope-intercept form shows the slopes are 2 and - 3 ⁄ 4 , respectively. The angle between two lines is given by
8+3 4 ------ 4– 6 4 ------ ------
2 3 4 –-- – 1 2 3 4 –-- + ------------
2 3 4 + -- 1 6 4 – -- ------
m 1 m 2 – 1 m 1 m 2 + ------------
11 2 ---
11 –2 ---
θ tan
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tan –1 11 2
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79.70 °
θ , by trigonometry. Distance Between a Point and a Line: The distance between a point ( x 1 , y 1 ) and a line given in the standard form Ax + By + C = 0 is d Ax 1 By 1 C + + A 2 B 2 + = ------------------- Example 9: Find the distance between the point (4,6) and the line 2 x + 3 y - 9 = 0. Solution: Using the formula:
Ax 1 By 1 C + + A 2 B 2 + -------------------
2 4 × 3 6 × – 9 + 2 2 3 2 + ----------------------
8+18– 9 4+9 --------------
17 13 -----
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d
Changing Coordinate Systems.— For simplicity it may be assumed that the origin in the Cartesian coordinate system coincides with the pole on a polar coordinate system and its x -axis with the polar horizontal axis. Then, if point P has polar coordinates of ( r , q ) and Cartesian coordinates of ( x , y ), by trigonometry x = r cos q and y = r sin q .
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