Equation of a Straight Line

• The angle which a line makes with the positive direction of x-axis measured in the anti-clockwise direction is called the inclination of the line.
• If ( 90°) is the inclination of a line, then tan is called its slope (or gradient).
• Slope of a line.
  (i) If the inclination of a line is , its slope = m = tan .
  (ii) Slope of a line through (x₁, y₁) and (x₂, y₂) is given by m =(y₂ -y_1)/(x₂ -x₁₎
• Equation of a straight line.
  (i) Equation of a line parallel to x-axis is y = b.
  (ii) Equation of a line parallel to y-axis is x = a.
  (iii) Equation of a line with slope m and y-intercept c is y = m x +c.
  (iv) Equation of a line through (x₁, y₁) and with slope m is y -y₁ = m (x -x₁).
• Conditions of parallelism and perpendicularity.
  Two lines with slopes m₁ and m₂ are
  (i) parallel if and only if m₁ = m₂.
  (ii) perpendicular if and only if m₁m₂ = -1.
• Reflection of P(, ) in the line y = x is P'(,).

Exercise

1. Find the inclination of a line whose gradient is
   (i) 1   (ii) 3   (iii) 1/3
2. Find the equation of a straight line parallel to x-axis and passing through the point (2, -7).
3. Find the equation of a straight line whose
   (i) gradient = 3, y-intercept = -4/3
   (ii) inclination = 30°, y-intercept = -3.
4. Write down the gradient and the intercept on the y-axis of the line 3y +2x = 12.
5. The equation to the line PQ is 3y -3x +7 = 0.
   (i) Write down the slope of the line PQ.
   (ii) Calculate the angle that the line PQ makes with the positive direction of x-axis.
6. The given figure represents the lines y = x +1 and y = 3 x -1. Write down the angles which the lines make with the positive direction of x-axis. Hence determine .
   [Hint. Ext. = sum of two opposite int. s; 60° = +45°.]
         
7. Given that (a, 2a) lies on the line y/2 = 3x -6, find the value of a.
8. The graph of the equation y = mx +c passes through the points (1, 4) and (-2, -5). Determine the values of m and c.
9. Find the equation of a straight line passing through (-1, 2) and having y-intercept 4 units.
10. Find the equation of a st. line whose inclination is 60° and passes through the point (0, -3).
11. Given that the line y/2 = x -p and the line ax +5=3y are parallel, find the value of a.
12. Find the value of m, if the lines represented by 2mx -3y = 1 and y = 1 -2x are perpendicular to each other.
13. If the lines 3x +y = 4, x -ay +7 = 0 and bx +2y +5 = 0 form three consecutive sides of a rectangle, find the values of a and b.
14. Find the equation of a straight line perpendicular to the line 2x +5y +7 = 0 and with y-intercept -3 units.
15. Find the equation of a straight line parallel to the line 2x +3y = 5 and having the same y-intercept as x +y +4 = 0.
16. Find the equation of the line which is parallel to 3x -2y -4 = 0 and passes through the point (0, 3).
17. Write down the equation of the line perpendicular to 3x +8y = 12 and passing through the point (-1, -2).
18. The co-ordinates of two points E and F are (0, 4) and (3, 7) respectively. Find
    (i) the gradient of EF
    (ii) the equation of EF
    (iii) the co-ordinates of the point where the line EF intersects the x-axis.
19. Find the equation of the line passing through the points (4, 0) and (0, 3). Find the value of k, if the line passes through (k,3/2).
20. If A(-1, 2), B(2, 1) and C(0, 4) are the vertices of a ABC, find the equation of the median through A.
21. Find the equation of a line passing through the point (-2, 3) and having x-intercept 4 units.
    [Hint. Since x-intercept is 4, the line passes through (4, 0).]
22. Find the equation of the st. line containing the point (3, 2) and making positive equal intercepts on the axes.
23. The intercepts made by a st. line on the axes are -3 and 2 units. Find
    (i) the gradient of the line.
    (ii) the equation of the line.
    (iii) the area of the triangle enclosed between the line and the co-ordinate axes.
24. A line through the point P (2, 3) meets the co-ordinate axes at points A and B. If PA = 2 PB, find the co-ordinates of A and B. Also find the equation of the line AB.
              
25. Calculate the co-ordinates of the point of intersection of the lines represented by x +y = 6 and 3x -y = 2.
26. The line joining the points P (4, k) and Q (-3, -4) meets the x-axis at A (1, 0) and y-axis at B. Find
    (i) the value of k.
    (ii) the ratio of PB : BQ.
27. Find the equations of the diagonals of a rectangle whose sides are x = -1, x = 2, y = -2 and y = 6.
28. Find the co-ordinates of the image of (3, 1) under reflection in x-axis followed by reflection in the line x =1.
29. If P'(-4, -3) is the image of the point P under reflection in the origin, find
    (i) the co-ordinates of P.
    (ii) the co-ordinates of the image of P under reflection in the line y = -2.
30. Find the co-ordinates of the image of the point P(4, 3) under reflection in the x-axis followed by reflection in the line x = -2.

Answers

1. (i) 45°        (ii) 60°       (iii) 30°.            2. y +7 = 0
3. (i) 33x -3y -4 = 0     (ii) x -3y -33 = 0
4. -2/3; 4                         5. (i) 1            (ii) 45°
6. 45°, 60°; 15°               7. 3
8. m = 3, c = 1                9. y = 2x +4
10.3x -y -3 = 0             11. 6
12. 3/4                              13. a = 3, b = 6
14. 5x -2y -6 = 0             15. 2x +3y +12 = 0
16. 3x -2y +6 = 0            17. 8x -3y +2 = 0
18. (i) 1        (ii) x -y +4 = 0     (iii) (-4, 0)
19. 3x +4y -12 = 0; 2      20. x -4y +9 = 0
21. x +2y -4 = 0              22. x +y -5 = 0
23. (i) 2/3        (ii) 2x -3y +6 = 0    (iii) 3 sq. units
24. A (6, 0), B (0, 9/2); 3x +4y -18 = 0
25. (2, 4)                           26. (i) 3 (ii) 4 : 3
27. 8x-3y+2 = 0, 8x+3y-10 = 0    28. (-1, -1)
29. (i) (4, 3)    (ii) (4, -7)                30. (-8, -3)