(1) Show that the lines are 3x + 2y + 9 = 0 and 12x + 8y − 15 = 0 are parallel lines. Solution
(2) Find the equation of the straight line parallel to 5x − 4y+3 = 0 and having x-intercept 3. Solution
(3) Find the distance between the line 4x + 3y + 4 = 0, and a point (i) (−2, 4) (ii) (7,−3) Solution
(4) Write the equation of the lines through the point (1,−1)
(i) parallel to x + 3y − 4 = 0
(ii) perpendicular to 3x + 4y = 6 Solution
(5) If (−4, 7) is one vertex of a rhombus and if the equation of one diagonal is 5x − y + 7 = 0, then find the equation of another diagonal. Solution
(6) Find the equation of the lines passing through the point of intersection lines 4x − y + 3 = 0 and 5x + 2y + 7 = 0, and (i) through the point (−1, 2) (ii) Parallel to x−y+5 = 0 (iii) Perpendicular to x − 2y + 1 = 0 Solution
(7) Find the equations of two straight lines which are parallel to the line 12x + 5y +2 = 0 and at a unit distance from the point (1, − 1). Solution
(8) Find the equations of straight lines which are perpendicular to the line 3x+ 4y −6 = 0 and are at a distance of 4 units from (2, 1). Solution
(9) Find the equation of a straight line parallel to 2x + 3y = 10 and which is such that the sum of its intercepts on the axes is 15 Solution
(10) Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from (−10,−2) to the line x + y − 2 = 0 Solution
(11) If p_{1} and p_{2} are the lengths of the perpendiculars from the origin to the straight lines x sec θ + y cosec θ = 2a and x cos θ − y sin θ = a cos 2θ, then prove that p_{1}^{2}+ p_{2}^{2}= a^{2}. Solution
(12) Find the distance between the parallel lines
(i) 12x + 5y = 7 and 12x + 5y+7 = 0
(ii) 3x − 4y + 5 = 0 and 6x − 8y − 15 = 0. Solution
(13) Find the family of straight lines (i) Perpendicular (ii) Parallel to 3x + 4y − 12 = 0. Solution
(14) If the line joining two points A(2,0) and B(3,1) is rotated about A in anticlockwise direction through an angle of 15°, then find the equation of the line in new position. Solution
(15) A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and it passes through the point (5,3). Find the co-ordinates of the point A. Solution
(16) A line is drawn perpendicular to 5x = y + 7. Find the equation of the line if the area of the triangle formed by this line with co-ordinate axes is 10 sq. units. Solution
(17) Find the image of the point (−2, 3) about the line x + 2y − 9 = 0. Solution
(18) A photocopy store charges Rs. 1.50 per copy for the first 10 copies and Rs. 1.00 per copy after the 10th copy. Let x be the number of copies, and let y be the total cost of photocopying. (i) Draw graph of the cost as x goes from 0 to 50 copies. (ii) Find the cost of making 40 copies Solution
(19) Find atleast two equations of the straight lines in the family of the lines y = 5x + b, for which b and the x-coordinate of the point of intersection of the lines with 3x − 4y = 6 are integers Solution
(20) Find all the equations of the straight lines in the family of the lines y = mx − 3, for which m and the x-coordinate of the point of intersection of the lines with x − y = 6 are integers. Solution
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