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- 1. ANALYTIC GEOMETRY (Lesson 6) Math 14 Plane and Analytic Geometry
- 2. STRAIGHT LINES/ FIRST DEGREE EQUATIONS
- 3. <ul><li>OBJECTIVES : </li></ul><ul><li>At the end of the lesson, the student is expected to be able to: </li></ul><ul><li>• Define and determine the general equation of a line </li></ul><ul><li>• Define and determine the different standard equations of line </li></ul><ul><li>Determine the directed distance from a point to a line </li></ul><ul><li>Determine the distance between parallel lines </li></ul>
- 4. STRAIGHT LINES A straight line is a locus of a point that moves in a plane with constant slope. It may also be referred to simply as a line which contains at least two distinct points. LINES PARALLEL TO A COORDINATE AXIS If a straight line is parallel to the y-axis, its equation is x = k, where k is the directed distance of the line from the y-axis. Similarly, if a line is parallel to the x-axis, its equation is y = k, where k is the directed distance of the line from the x-axis.
- 6. DIFFERENT STANDARD FORMS OF THE EQUATION OF A STRAIGHT LINE A. POINT-SLOPE FORM : If the line passes through the point (x 1 , y 1 ), then the slope of the line is . Rewriting the equation we have which is the standard equation of the point-slope form.
- 7. The equation of the line through a given point P 1 (x 1 , y 1 ) whose slope is m. y x
- 8. <ul><li>EXAMPLE : </li></ul><ul><li>Find the general equation of the line: </li></ul><ul><li>through (2,-7) with slope of 2/5 </li></ul><ul><li>through the point (-3, 4) with slope of -2/5 </li></ul>B. TWO-POINT FORM : If the line passes through the points (x 1 , y 1 ) and (x 2 , y 2 ), then the slope of the line is . Substituting it in the point-slope formula, we have which the standard equation of the two-point form.
- 9. The equation of the line through points P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) y x
- 10. <ul><li>EXAMPLE : </li></ul><ul><li>Find the general equation of the line: </li></ul><ul><li>passing through (4,-5) and (-6, 3) </li></ul><ul><li>passing through (2,-3) and (-4, 5) </li></ul>C. SLOPE-INTERCEPT FORM : Consider a line not parallel to either axes of the coordinate axes. Let the slope of the line be m and intersecting the y-axis at point (0, b), then the slope of the line is . Rewriting the equation, we have which is the standard equation of the slope-intercept form.
- 11. The equation of the line having the slope, m, and y-intercept (0, b) y x
- 12. <ul><li>EXAMPLE : </li></ul><ul><li>Find the general equation of the line with slope 3 and y-intercept of 2/3. </li></ul><ul><li>Express the equation 3x-4y+8=0 to the slope-intercept form and draw the line. </li></ul>D. INTERCEPT FORM : Let the intercepts of the line be the points (a, 0) and (0, b). Then the slope of the line and its equation is . Simplifying the equation we have which is the standard equation of the intercept form.
- 13. The equation of the line whose x and y intercepts are (a, 0) and (0, b) respectively. y x
- 14. <ul><li>EXAMPLE : </li></ul><ul><li>Find the general equation of the line: </li></ul><ul><li>with x-intercept of 2 and y-intercept of -3/4 </li></ul><ul><li>through (-2, 7) with intercepts numerically equal but of opposite sign </li></ul><ul><li>NORMAL FORM : </li></ul><ul><li>Suppose a line L , whose equation is to be found, has its distance from the origin to be equal to p . Let the angle of inclination of p be </li></ul>
- 16. Since p is perpendicular to L , the slope of p is equal to the negative reciprocal of the slope L . Substituting in the slope-intercept form, y = mx + b , we obtain Simplifying, we have the normal form of the straight line
- 17. Reduction of the General Form to the Normal Form The slope of the line Ax+By+C=0 is . The slope of p which is perpendicular to the line is therefore . Thus, . From Trigonometry, we obtain the values and . If we divide through the general equation of the straight line by , we have Transposing the constant to the right, we obtain This is of the normal form . Comparing the two equations, we note that .
- 18. <ul><li>EXAMPLE : </li></ul><ul><li>Reduce 5x+3y-4=0 to the normal form. </li></ul><ul><li>2. Find the equation of a line parallel to the line 4x-y+8=0 passing at a distance ±3 from the point (-2,-4). </li></ul>
- 20. PARALLEL AND PERPENDICULAR LINES The lines Ax+By+C=0 and Ax+By+K=0 are parallel lines. But, the lines Ax+By+C=0 and Bx-Ay+K=0 are perpendicular lines. <ul><li>EXAMPLE : Find the general equation of the line: </li></ul><ul><li>through (-3, 8) parallel to the line 6x-5y+15=0 </li></ul><ul><li>through (6,-1) and perpendicular to the line 4x-5y-6=0 </li></ul><ul><li>passing through (-1, 5) and parallel to the line through (1 ,3) and (1,-4) </li></ul>
- 21. DIRECTED DISTANCE FROM A POINT TO A LINE The directed distance from the point P(x 1 , y 1 ) to the line Ax+By+C=0 is , where the sign of B is Taken into consideration for the sign of the . If B>0, then it is and B<0, then it is . But if B=0, take the sign of A.
- 22. y x
- 23. <ul><li>EXAMPLE : </li></ul><ul><li>Find the distance of the point (6,-3) from the line 2x-y+4=0. </li></ul><ul><li>Find the equation of the bisector of the acute angle for the pair of lines L 1 : 11x+2y-7=0 and L 2 : x+2y+2=0. </li></ul><ul><li>Find the distance between the lines 3x+y-12=0 and 3x+y-4=0 </li></ul>
- 24. EXERCISES : 1. Determine the equation of the line passing through (2, -3) and parallel to the line passing through (4,1) and (-2,2). 2. Find the equation of the line passing through point (-2,3) and perpendicular to the line 2x – 3y + 6 = 0 3. Find the equation of the line, which is the perpendicular bisector of the segment connecting points (-1,-2) and (7,4). 4. Find the equation of the line whose slope is 4 and passing through the point of intersection of lines x + 6y – 4 = 0 and 3x – 4y + 2 = 0
- 25. 5. The points A(0, 0), B(6, 0) and C(4, 4) are vertices of triangles. Find: a. the equations of the medians and their intersection point b. the equations of the altitude and their intersection point c. the equation of the perpendicular bisectors of the sides and their intersection points
- 26. Exercises: 1. Find the distance from the line 5x = 2y + 6 to the points a. (3, -5) b. (-4, 1) c. (9, 10) 2. Find the equation of the bisector of the pair of acute angles formed by the lines 4x + 2y = 9 and 2x – y = 8. 3. Find the equation of the bisector of the acute angles and also the bisector of the obtuse angles formed by the lines x + 2y – 3 = 0 and 2x + y – 4 = 0.
- 28. REFERENCES Analytic Geometry, 6 th Edition, by Douglas F. Riddle Analytic Geometry, 7 th Edition, by Gordon Fuller/Dalton Tarwater Analytic Geometry, by Quirino and Mijares

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