COORDINATE GEOMETRY

Chapter objectives

By the end of this chapter, you should be able to:

1. derive the general equation of a straight line.

2. determine the coordinates of mid-point of a line segment.

3. calculate the distance between two points on the xy-plane.

4. determine the conditions for both parallel and perpendicular lines.
5. solve problems involving parallel and perpendicular lines.

6.  solve problems on parallel and perpendicular lines in real-life situation.

Equation of a straight line

Consider two points (-2 , 1) and (2 , 3) as shown in Figure 1 .1 below:

Figure 1. 1


The gradient (m) of the line through these points is given by;



If (x , y) is another point on the line, the gradient m is also given by But, the gradient of a straight line is constant at any point on the straight line.Therefore,

 


From the equation above, the gradient of the line is a coefficient of x, in this case and the y-intercept is 2.

In general, if a gradient of a line is m and the y-intercept is c, its equation is y = mx + c.

Consider two known points (Xv 1/1) and (xy !/2) on a straight line with an unknown point (x, y) as shown in Figure 1 .2 below.

Figure 1.2

The gradient m of the line is given by:

              

Worked Example 1

Find the equation of a straight line which passes through (2 , 3) and whose slope is 2/3

Example 1.2

Find the equation of a straight line passing through the points (0 , -1) and (2 , 4) in the form of ax + by + c = O.


Worked example 2

Worked Example 3

Revision Exercise 1


  1.     Calculate the equation of a straight line passing through (2 , -4) which has the same slope as that of the line 3x — 6y + 12 = 0.
  2.  Find the equation of a straight line passing through (3 , 3) whose slope is the same as the slope of the line joining (3 , 3) to (2 , 4).
  3.  Calculate the equation of a straight line with slope — having the same y-intercept as the line — + 1 0 = O.
  4. Find the equation of a straight line through the point (5 , 0) with infinite gradient.
  5. Find the equation of a straight line through the point (0 , 4) with gradient zero.
  6. Find the equation of a straight line passing through the point of intersection of the lines:

11 : 3y + 3 = 2x, 12 2+ 2 = 3x having the same gradient as the line 3x— 6y + 18 = 0, in the form of ax + by + c = 0

vii. Find the equation of a straight line passing through (3 , 4) and its slope is 1.

viii.        Find the equation of a straight line shown in the figure below in the form of y = mx + c.

ix.    Find the equation of a straight line with x-intercept -2 and y-intercept -3.


Mid-point of a line segment

The mid-point M of a line segment AB is the point which divides the line segment into halves. Thus, ÄM = MB

Consider two points A (xl , yo and B , y2) on the same straight line on the xy- plane as shown in Figure 1.3 below:

Figure 1.3

Figure 1.4



Worked example 4

A parallelogram has vertices A (1 , 7), B (6 , 4), C (3 , 1) and D (-2 , 4).

(a)        Find the coordinates of the mid-point of the diagonals.

(b)       Show that the diagonals bisect each other.

Solution

Exercise 2


 1.   Given the end points of     line segments, find the mid-point of each line segment.

       (a)          (3 , 4) and (4 ,-4)                                 (b)      (-7 , 12) and (3 , -8)

(c)                (0 , -3) and (-1 , -1)                           

         (d)        (3 , 5) and (-5 , -3)                                     

2.      M is the mid-point of the line segment AB. If the coordinates of M are (2 , 1) and the coordinate of A are (-3 , 1), find the coordinates of B.

3. A straight line is defined by the equation x + y = 12 and crosses the y-axis at A and meets the line x = 6 at B. If C is the point (0 , 3) and M is the mid-point of n, find: (a) the coordinates of A, B and M

(b)     the equation of line CM

(c)     the equation of the line passing through M whose slope is 4.

4. If the points (1 , 2), (1 , -3), (-4 , -3) and (-4 , 3) are the vertices of a square, show that the diagonals bisect each other.

5.The coordinates of P and Q of a parallelogram PQRS are (0 , O) and (-4 , 6) respectively. The diagonal intersect at (4 , 8). Find the coordinates of R and S.

Distance between two points on a plane

Consider two points A (Xl , yo and B (x2 , Y2) on a plane as shown in Figure below; To find the distance between two points on the plane, draw a right angled triangle ABC with line segment AB as the hypotenuse as shown below.

Figure 1.5

The coordinates of C are (x2 , yo. By using Pythagoras Theorem,

Worked Example 4

Find the distance between the points A (-1, -1) and B (2 , 3) on a plane.

Solution

        Let x1 = -1, y1=-1, x2 = 2 and y2 = 3

The distance d between points A and B is given by

d = 5 units long if the order is reversed, still the answer is the same:

d = 5 units long therefore,distance between A (-1 -1) and B (2 , 3) is 5 units long.

worked example 5

The distance between points P (k , 2) and Q (-3 , 5) is 5. Find the possible values of k. Solution


Exercise 3

1. Find the distance between the given lines of a segment
(a) (-2,-3) and (4,5)
(b) (5,4) and (8,-2)
(c) (a,b) and -2a,3b)
(d) (1/2,1/3) and 1/4,1/5)
2. Given that the distance from point A (-2,2) TO POINT B whose abscisca is 13 is 17, find the ordinate of the point B
3.Prove that a triangle whose vertices are A (1 , -1), B (4 , -1) and C (1 , 3) is a right triangled triangle.
4. Show that, the point (-3 , -4) is equidistant from the points (4 , -8) and (-2 , 4).
5. Given a triangle with vertices A (0 , -2) , B (4 , -6) and C (3/2,-9/2)  , show that the
 
triangle is an isosceles.
6. Find the distance of a line segment with end points (6 , 7) and (-2 , 1).
7.Show that the triangle with vertices A (0 , 0) , B (14 , 2) and C (-6 , 8) is an isosceles triangle.
8. Find the distance of the point (8 , 6) from the origin in the xy-plane.
9.The vertices of a parallelogram are A (2 , 1), B (6 , 1), D (5 , -2) and C (1 , -2). Show that the diagonals are not equal.

Parallel lines

Two or more lines which do not meet however long they are extended and the distance, between them is constant are called parallel lines. Railway lines are good examples of parallel lines.

Figure 1.6

From points R and S, lines RQ and ST are drawn perpendicular to PT.

From Figure 1 .6, A PRQ is similar to A QST. (AA-Similarity theorem.)

Slope of L1 = RQ/PQ and that of L2 = ST/QT


But,  (Ratios of corresponding sides of similar triangles.)

Since  is slope (ml) of the line L1 and  ST/QT is slope (m2) of the line l2

Therefore, parallel lines have equal slopes.

Worked Example 6

Find the equation of a straight line passing through the origin and parallel to the line 4x — 3y— 2 = O. Write it in the form of ax + by + c = 0.

Solution

Exercise 4

  1. Find the equation of a straight line passing through the point (3 , 4) and parallel to the line whose equation is 12x 6y = 9.
  2. Show that the line of the equation 3x y + 6 = 0 is parallel to the line of the equation 6x -2y + 7 = 0.
  3. The line passing through the point A (k,4) and B (3,2k) is parallel to the line

y + 3x — 4 = 0. Find the value of k.                                                                                                

4.        What value of will make the line containing the point (k , 5) and (-2 , 3)        parallel to the line containing the point (6 k) and (2 , 0)?

5.  Find the equation of a straight line passing through (-4 , 3) and parallel to

2x/ _ 3Y/ =1

3      4

 6. Find the equation of a straight line in the form of ax + by + c = 0 which passes

through (6 , 1) and parallel to 2x — y + 7 = 0.

7.     Find the equation of a straight line which is parallel to  and passes

through (c,-1/c)

8.   If the lines of the equations 2y — 4x = 3 and Ax + y = 4 are parallel, find the value of A.

9.    Find the equation of a straight line passing through (-2, 3) and parallel to the line passing through (3, 2) and (-1, 6).

10.   Find the equation of a straight line passing through the point of intersection of the lines 2x + = 3 and 3x — 2y = 8 and the point (3, 2).

11.    Are the running tracks parallel? Explain.


Perpendicular lines

Two lines are perpendicular (normal) to each other if they meet at right angle. Consider two perpendicular lines Ll and L2 with slopes

Figure 1.7

From Figure 1 .7 above, angles a and b, a and c are pairs of complementary angles, therefore, angle b = angle c and A ADB A BFC. If a + b = 900 and a + c = 900 then, b = c.

 

Worked example 7

Find the equation of a straight line passing through (-4 , 2) and perpendicular to line whose equation is x + 2y = 4.

Solution

From the equation x + 2y = 4, make y the subject of the formula.

             

Worked example 7

Find the equation of a straight line which is a perpendicular bisector to the line segment whose end points are A (9 , 1) and B (7 , -5).

Solution

Exercise 5

1.       Find the equation of a perpendicular bisector to the line segment whose end-points have coordinates (0 , 2) and (3 , 6).

2.              What value of t will make the line passing through the point 5 , t) and B (1 , 2) perpendicular to the line passing through the points C (-1 , -2) and

3.              Determine the coordinates of the point P (x , y) such that the line joining it to the point (3 , -1) forms a right angle with the line through the points (3 , -1) and (-5 , -5).

4.              Line I is perpendicular to the line joining the points ( 3 , 2) and (5 , 6). If it passes through the point of intersection of the lines 2x —y = 1 and 3x + 3y = 6, determine the equation of the line l.

5.              Find an equation of the perpendicular bisector of the line joining A (-2 , 4) and

6.              Find the equation of a straight line passing through (3 , -2) and perpendicular to the line which passes through the points ( 4 , 3) and (2 , 6).

7.              What are the coordinates of the foot of the perpendicular from (2 , 3) to the line

8.              Find the equation of a straight line passing through (a , -2a) and perpendicular to the line ay+x= 1 .

                                                                                                                  

9.              Find the equation of a straight line perpendicular to the line        and

              passing through the x-intercept of the line

10.          Find the value of t so that the point P (-2, t) is on the line which is the perpendicular bisector of the line segment joining the coordinates A (4, -1) and B (8, -7).

  1 1 .  The vertices of triangle ABC are A (4 , -2), B (2 , 1), C (-3 , -3). Find:

(a) an equation of the perpendicular bisector of the side AC. (b) an equation of the perpendicular to the side BC.



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