1Which is the formula for the angle between the straight lines \( y = m_1x + c \) and \( y = m_2x + c \): \( \theta = \tan^{-1}\pm\left(\frac{m_1 + m_2}{1 - m_1m_2}\right) \), \( \theta = \tan^{-1}\pm\left(\frac{1 - m_1m_2}{m_1 - m_2}\right) \), \( \theta = \tan^{-1}\pm\left(\frac{m_1 - m_2}{1 + m_1m_2}\right) \) or \( \theta = \tan^{-1}\pm\left(\frac{1 + m_1m_2}{m_1 - m_2}\right) \)?
2Which is the condition for two straight lines to be parallel: \( m_1 - m_2 = 1 \), \( m_1m_2 = -1 \), \( m_1 + m_2 = -1 \) or \( m_1 - m_2 = 0 \)?
3Which is the condition for two straight lines to be perpendicular: \( m_1m_2 = 1 \), \( m_1m_2 = -1 \), \( m_1 + m_2 = -1 \) or \( m_1 - m_2 = 0 \)?
4What is the angle between the lines \( 3x - y - 7 = 0 \) and \( x + 3y - 6 = 0 \): 0°, 30°, 45° or 90°?
5What is the acute angle between the lines \( x - 3y - 2 = 0 \) and \( 2x - y - 1 = 0 \): 0°, 30°, 45° or 60°?
6What is the obtuse angle between the lines \( \sqrt{3}x - y - 5 = 0 \) and \( y + 10 = 0 \): 120°, 135°, 150° or 160°?
7Find the acute angle between the lines \( y - 3x - 4 = 0 \) and \( y = -2x + 1 \).
8Find the acute angle between the lines \( y = 3x + 2 \) and \( x + 2y - 2 = 0 \).
9Find the obtuse angle between the lines \( y = 3x + 4 \) and \( y = -2x + 1 \).
10Find the obtuse angle between the lines \( 2x + 3y - 4 = 0 \) and \( x + 2y - 3 = 0 \).
11Find the angles between the lines \( 3x + y - 2 = 0 \) and \( x + 3y + 12 = 0 \).
12Find the angles between the lines \( 2x - y + 3 = 0 \) and \( y = -\frac{1}{2}x - 4 \).
13Prove that the lines \( 2x - 3y - 5 = 0 \) and \( 2x - 3y = 12 \) are parallel.
14Prove that the lines \( x - 5y = 4 \) and \( 3x - 15y + 12 = 0 \) are parallel.
15Prove that the lines \( y + \sqrt{3}x + 4 = 0 \) and \( x - \sqrt{3}y = 5 \) are perpendicular.
16Prove that the lines \( ax + by + c = 0 \) and \( bx - ay + c = 0 \) are perpendicular.
17The lines \( px + 3y - 12 = 0 \) and \( 4y - 3x + 7 = 0 \) are parallel. Find p.
18The lines \( 2x - 8y + 6 = 0 \) and \( qx - 12y - 4 = 3 \) are parallel. Find q.
19The lines \( ax + 3y - 4 = 0 \) and \( 3x + 9y - 5 = 0 \) are parallel. Find a.
20The lines \( 3x + my = 5 \) and \( \frac{x}{2} + \frac{y}{3} = 1 \) are parallel. Find m.
21The lines \( px + 3y - 12 = 0 \) and \( 4y - 3x + 7 = 0 \) are perpendicular. Find p.
22The lines \( 2x + 3y + 6 = 0 \) and \( mx - 5y + 20 = 0 \) are perpendicular. Find m.
23The lines \( 4x + 3y - 7 = 0 \) and \( 3x + qy - 5 = 0 \) are perpendicular. Find q.
24The lines \( ax + 5y - 16 = 0 \) and \( 6x + 10y - 9 = 0 \) are perpendicular. Find a.
25For the lines \( px + qy + r = 0 \) and \( lx - my + n = 0 \), find the slope of the line \( px + qy + r = 0 \).
26If the lines \( px + qy + r = 0 \) and \( lx - my + n = 0 \) are perpendicular, prove that \( pl = qm \).
27Find the equation of the line that passes through (3, 4) and is parallel to the line \( 3x + 4y - 12 = 0 \).
28Find the equation of the line that is parallel to the line joining (2, 3) and (3, \( -1 \)) and passes through (3, 1).
29Find the equation of the line that passes through (2, 5) and is perpendicular to the line \( 2x + 5y + 31 = 0 \).
30Find the equation of the line that is perpendicular to the line joining (4, 7) and (2, \( -3 \)) and passes through (3, 1).
31The lines \( 3x + 4y - 7 = 0 \) and \( 5x - 2y - 3 = 0 \) meet at Z. Find the coordinates of Z.
32Find the equation of the straight line through Z, the meeting point of \( 3x + 4y - 7 = 0 \) and \( 5x - 2y - 3 = 0 \), that is perpendicular to \( 2x + 3y = 5 \).
33The line \( \frac{x}{a} + \frac{y}{b} = 1 \) passes through the meeting point of \( x + y - 3 = 0 \) and \( 2x - 3y - 1 = 0 \), and is parallel to \( x - y - 6 = 0 \). Find a and b.
34Find the equation of the perpendicular bisector of the line joining (\( -2 \), 3) and (4, 7).
35Find the equation of the perpendicular bisector of the line joining (4, \( -2 \)) and (\( -8 \), 9).
36Find the equation of the perpendicular bisector of the line joining (6, \( -5 \)) and (\( -8 \), 9).
37Find the equations of the two lines through (3, \( -2 \)) that make an angle of 45° with the line \( 2x - 3y + 10 = 0 \).
38Find the equations of the two lines through (0, 0) that make an angle of 60° with the line \( x + y + 3 = 0 \).
39Why does the formula for the angle between two lines carry a \( \pm \) sign?
40Find the acute and obtuse angles between the lines \( y = \sqrt{3}x + 1 \) and \( y = -\sqrt{3}x + 4 \).
41Show that the line through (1, 2) and (3, 6) is parallel to the line \( 2x - y + 7 = 0 \).
42The lines x = 4 and y = 2 are perpendicular. Why can the condition \( m_1m_2 = -1 \) not be used to show this?