ANALYTIC GEOMETRY : LINES

LINEAR EQUATIONS

The equation of a line can be written in the form

Ax + By + C = 0

where A, B and C are constants. This is called the general equation of a line. If A = 0 and B and C are not zero, the line is horizontal. If B = 0 and A and C are not zero, the line is vertical. If C = 0 and A and B are not zero, the line passes through the origin or the point (0, 0).


The Standard Equation of Lines

1. The point-slope form

y – y₁ = m ( x – x₁)

where m is a slope and (x₁, y₁) is a given point

2. The slope-intercept form

y = m x + b

where m is a slope and b is the y – intercept

3. The two-point form

y – y₁ = [(y₂ – y₁)/(x₂ – x₁)] (x – x₁)

where (x₁, y₁) and (x₂, y₂) are given points.

4. The intercept form

x/a + y/b = 1

where a is an x – intercept and b is the y–intercept.

5. The Normal form

x cos a + y sin a = p

where a is the slope of the normal and p is the distance of the
line from the origin.



Exercises (Problems in red color are assignment)

I. Find the equation of the line given the conditions:
1. P( 3 , – 2) , m = ¾
2. P(0, 5 ), m = – 2/3
3. P₁(-1, 2 ) , P₂(3, 4 )
4. P₁( 0, -3 ), P₂( 4, 0 )
5. m = ½ , b = -2
6. m = 3, b = - 3/2
7. a = 60 degrees, p = 3
8. a = 45 degrees, p = 3sqrt(2)
9. a = - ½ , b = 2
10. a = 3 , b = –5

II. Find the intersection of the following lines.
1.   x + 2y = 3
      2x + 8 = 3y

2.   x – 2y = 2
      2x – 3y = 5

3.   3x – y = 2
       x + 2y = 3

4.   x – y = 2
      5x + 3y = 2

5.   3x - 2y = 4
      2x + y = 12

III. The vertices of the triangle MNO are M(-1, 1), N(6, 2 ) and O(2, 5 ).

1. Determine the equation of (a) the side MN, (b) the medial from M
to NO and ( c ) the altitude from M to NO.
Ans. 7y – x = 8, 2y – x = 3 , 3y – 4x = 7

2. Find the equation of ( a ) the line through N parallel to MO,
( b) the line through O parallel to MN and ( c) the point of
intersection of the lines in ( a) and ( b).
Ans. 4x – 3y = 18, 7y – x = 33 , ( 9 , 6 )

3. The angle between MN and NO and between MN and MO.

IV.   The equation of two sides of a parallelogram 2x – 3y + 7 = 0
and 4x + y = 21, and one of the vertex is (–1, –3 ).
Determine the three other vertices.
Ans. (5 , 1) , ( 4 , 5), (– 2, 1)