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EQUATIONS OF A LINE in .NET
EQUATIONS OF A LINE Scan QR Code In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. Denso QR Bar Code Drawer In .NET Framework Using Barcode drawer for VS .NET Control to generate, create QR image in Visual Studio .NET applications. Consider a line L that passes through the point P1 (x1 , y1 ) and has slope m [Fig. 46(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) QR Decoder In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. Draw Bar Code In .NET Using Barcode generator for .NET Control to generate, create bar code image in Visual Studio .NET applications. STRAIGHT LINES
Scanning Barcode In Visual Studio .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Create QR Code In C#.NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications. [CHAP. 4
QR Code Creation In .NET Framework Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. QRCode Encoder In Visual Basic .NET Using Barcode drawer for .NET Control to generate, create QR Code image in .NET applications. Fig. 44 Generate Bar Code In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. Code 128 Code Set C Encoder In VS .NET Using Barcode maker for .NET framework Control to generate, create Code 128A image in .NET applications. Fig. 45 On the other hand, if P(x, y) is not on line L [Fig. 46(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) Barcode Generator In .NET Framework Using Barcode creation for .NET framework Control to generate, create barcode image in Visual Studio .NET applications. MSI Plessey Printer In .NET Using Barcode maker for .NET framework Control to generate, create MSI Plessey image in .NET framework applications. 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 pointslope equation of the line L . EXAMPLES Data Matrix 2d Barcode Printer In None Using Barcode drawer for Office Excel Control to generate, create Data Matrix 2d barcode image in Office Excel applications. Making EAN13 In Visual Studio .NET Using Barcode generation for Reporting Service Control to generate, create EAN 13 image in Reporting Service applications. (a) A pointslope equation of the line going through the point (1, 3) with slope 5 is y 3 = 5(x 1) Drawing Code 128C In Java Using Barcode creation for Java Control to generate, create Code128 image in Java applications. ECC200 Maker In None Using Barcode printer for Software Control to generate, create Data Matrix ECC200 image in Software applications. CHAP. 4] ECC200 Generation In Java Using Barcode drawer for Android Control to generate, create Data Matrix image in Android applications. Make Barcode In Java Using Barcode encoder for Java Control to generate, create barcode image in Java applications. STRAIGHT LINES
Barcode Creator In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. Drawing Data Matrix 2d Barcode In ObjectiveC Using Barcode generation for iPhone Control to generate, create Data Matrix image in iPhone applications. Fig. 46 (b) Let L be the line through the points (1, 4) and ( 1, 2). The slope of L is m= Therefore, two pointslope 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 ycoordinate of the point where L intersects the yaxis (see Fig. 47). The number b is called the yintercept of L , and (4.3) is called the slopeintercept equation of L . Fig. 47 STRAIGHT LINES
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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 slopeintercept 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 slopeintercept equation. An alternative method is to write down a pointslope 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 yaxis [see Fig. 48(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 (anglesideangle) 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. 48 Hence, R1 Q1 = R2 Q2 , and slope of L1 = Thus, parallel lines have equal slopes. R1 Q1 R2 Q2 = = slope of L2 1 1 CHAP. 4] STRAIGHT LINES
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. 48(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 slopeintercept equation y = 3x 2. Hence, the slope of M is 3, and the slope of the parallel line L also must be 3. The slopeintercept 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 slopeintercept 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.

