EQUATIONS OF A LINE in .NET

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EQUATIONS OF A LINE
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Consider a line L that passes through the point P1 (x1 , y1 ) and has slope m [Fig. 4-6(a)]. For any other point P(x, y) on the line, the slope m is, by de nition, the ratio of y y1 to x x1 . Hence, y y1 =m x x1 (4.1)
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Fig. 4-5 On the other hand, if P(x, y) is not on line L [Fig. 4-6(b)], then the slope (y y1 )/(x x1 ) of the line PP1 is different from the slope m of L , so that (4.1) does not hold. Equation (4.1) can be rewritten as y y1 = m(x x1 ) (4.2)
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Note that (4.2) is also satis ed by the point (x1 , y1 ). So a point (x, y) is on line L if and only if it satis es (4.2); that is, L is the graph of (4.2). Equation (4.2) is called a point-slope equation of the line L . EXAMPLES
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(a) A point-slope equation of the line going through the point (1, 3) with slope 5 is y 3 = 5(x 1)
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Fig. 4-6
(b) Let L be the line through the points (1, 4) and ( 1, 2). The slope of L is m= Therefore, two point-slope equations of L are y 4=x 1 and y 2=x+1 4 2 2 = =1 1 ( 1) 2
Equation (4.2) is equivalent to y y1 = mx mx1 or y = mx + (y1 mx1 )
Let b stand for the number y1 mx1 . Then the equation becomes y = mx + b (4.3)
When x = 0, (4.3) yields the value y = b. Hence, the point (0, b) lies on L . Thus, b is the y-coordinate of the point where L intersects the y-axis (see Fig. 4-7). The number b is called the y-intercept of L , and (4.3) is called the slope-intercept equation of L .
Fig. 4-7
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EXAMPLE Let L be the line through points (1, 3) and (2, 5). Its slope m is
2 5 3 = =2 2 1 1 Its slope-intercept equation must have the form y = 2x + b. Since the point (1, 3) is on line L , (1, 3) must satisfy the equation 3 = 2(1) + b So, b = 1, giving y = 2x + 1 as the slope-intercept equation. An alternative method is to write down a point-slope equation, y 3 = 2(x 1) whence, y 3 = 2x 2 y = 2x + 1
PARALLEL LINES
Assume that L1 and L2 are parallel, nonvertical lines, and let P1 and P2 be the points where L1 and L2 cut the y-axis [see Fig. 4-8(a)]. Let R1 be one unit to the right of P1 , and R2 one unit to the right of P2 . Let Q1 and Q2 be the intersections of the vertical lines through R1 and R2 with L1 and L2 . Now P1 R1 Q1 is congruent to P2 R2 Q2 .
geometry Use the ASA (angle-side-angle) congruence theorem. R1 = R2 since both are right angles, P1 R1 = P2 R2 = 1
P1 = P2 , since P1 and P2 are formed by pairs of parallel lines.
Fig. 4-8 Hence, R1 Q1 = R2 Q2 , and slope of L1 = Thus, parallel lines have equal slopes. R1 Q1 R2 Q2 = = slope of L2 1 1
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Conversely, if different lines L1 and L2 are not parallel, then their slopes must be different. For if L1 and L2 meet at the point P [see Fig. 4-8(b)] and if their slopes are the same, then L1 and L2 would have to be the same line. Thus, we have proved: Theorem 4.1: Two distinct lines are parallel if and only if their slopes are equal. EXAMPLE Let us nd an equation of the line L through (3, 2) and parallel to the line M having the equation 3x y = 2. The line M has slope-intercept equation y = 3x 2. Hence, the slope of M is 3, and the slope of the parallel line L also must be 3. The slope-intercept equation of L must then be of the form y = 3x + b. Since (3, 2) lies on L , 2 = 3(3) + b, or b = 7. Thus, the slope-intercept equation of L is y = 3x 7. An equivalent equation is 3x y = 7. 4.4 PERPENDICULAR LINES
Theorem 4.2: Two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Hence, if the slope of one of the lines is m, then the slope of the other line is the negative reciprocal 1/m. For a proof, see Problem 4.5.
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