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.
5.
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.
9. The equation of the line through points P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) y x
10.
11. The equation of the line having the slope, m, and y-intercept (0, b) y x
12.
13. The equation of the line whose x and y intercepts are (a, 0) and (0, b) respectively. y x
14.
15.
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.
19.
20.
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.
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.
27.
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
