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# Equations of Straight Lines

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### Equations of Straight Lines

1. 1. T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
2. 2. Point-slope form x Suppose that Q (x0, y0) is a fixed point on a non-vertical line L, whose slope is m. Let P (x, y) be an arbitrary point on L. Then, by the definition, the slope of L is given by Since the point Q (x0, y0) along with all points (x, y) on L satisfies (1) and no other point in the plane satisfies (1). Equation (1) is indeed the equation for the given line L. Thus, the point (x, y) lies on the line with slope m through the fixed point (x0, y0), if and only if, its coordinates satisfy the equation Y Q (x0 ,y0) P (x , y) o L Slope = m 0 0 xx yy m )( 00 xxmyyor --------------- (1) )( 00 xxmyy © iTutor. 2000-2013. All Rights Reserved
3. 3. Two-point form  Let the line L passes through two given points A (x1, y1) and B (x2, y2).  Let P (x, y) be a general point on L  The three points A, B and P are collinear, therefore, we have slope of AP = slope of BP i.e., Thus, equation of the line passing through the points A (x1, y1) and B (x2, y2) is given by x Y A (x1 ,y1) P (x , y) o L B (x2 ,y2) 12 12 1 1 xx yy xx yy or 1 yy 1 12 12 1 xx xx yy yy 1 12 12 1 xx xx yy yy © iTutor. 2000-2013. All Rights Reserved
4. 4. Slope-intercept form x Y o L • Sometimes a line is known to us with its slope and an intercept on one of the axes. We will now find equations of such lines Case I Suppose a line L with slope m cuts the y-axis at a distance c from the origin. The distance c is called they-intercept of the line L. Obviously, coordinates of the point where the line meet the y- axis are (0, c). Thus, L has slope m and passes through a fixed point (0, c). Therefore, by point-slope form, the equation of L is y – c = m(x – 0 ) y =mx + c (0 ,c) © iTutor. 2000-2013. All Rights Reserved
5. 5. Thus, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if. y =mx + c Case II Suppose line L with slope m makes x-intercept d. Then equation of L is y = m(x – d) © iTutor. 2000-2013. All Rights Reserved
6. 6. Intercept - form  Suppose a line L makes x-intercept a and y-intercept b on the axes. Obviously L meets x-axis at the point (a, 0) and y-axis at the point (0, b). By two-point form of the equation of the line, we have Or i.e., Thus, equation of the line making intercepts a and b on x- and y-axis, respectively, is x Y (0 , b) o L (a , 0) b a ax a b y 0 0 0 abbxay 1 b y a x 1 b y a x © iTutor. 2000-2013. All Rights Reserved
7. 7. Normal form Suppose a non-vertical line is known to us with following data: a) Length of the perpendicular (normal) from origin to the line. b) Angle which normal makes with the positive direction of x- axis. Let L be the line, whose perpendicular distance from origin O be OA = p and the angle between the positive x-axis and OA be ∠ XOA = . The possible positions of line L in the Cartesian plane. Now, our purpose is to find slope of L and a point on it. Draw perpendicular AM on the x-axis in each case. © iTutor. 2000-2013. All Rights Reserved
8. 8. x x x x X’ X’ X’ X’ Y Y Y Y Y’ Y’Y’ Y’ L L L L A A A A M M MM pp pp oo oo (iv) (iii) (ii)(i) © iTutor. 2000-2013. All Rights Reserved
9. 9. In each case,  we have OM = p cos and MA = p sin , so that the coordinates of the point A are (pcos , psin ) Further, line L is perpendicular to OA. Therefore, The slope of the line L = Thus, the line L has slope and point A( pcos , psin ) on it. Therefore, by point-slope form, the equation of the line L is sin cos tan 11 slopeofOA sin cos cos sin cos sin pxpy © iTutor. 2000-2013. All Rights Reserved
10. 10. Or Hence, the equation of the line having normal distance p from the origin and angle which the normal makes with the positive direction of x-axis is given by pyx sincos pyx sincos © iTutor. 2000-2013. All Rights Reserved
11. 11. The End Call us for more Information: www.iTutor.com Visit 1-855-694-8886