(1) Find the combined equation of the straight lines whose separate equations are x − 2y −3 = 0 and x + y+5 = 0. Solution
(2) Show that 4x^{2} + 4xy + y^{2} − 6x − 3y − 4 = 0 represents a pair of parallel lines. Solution
(3) Show that 2x^{2} + 3xy − 2y^{2} + 3x + y + 1 = 0 represents a pair of perpendicular lines. Solution
(4) Show that the equation 2x^{2} −xy−3y^{2} −6x + 19y − 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is tan^{−1}(5). Solution
(5) Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x^{2} − 2xy sec 2α + y^{2} = 0 Solution
(6) Find the equation of the pair of straight lines passing through the point (1, 3) and perpendicular to the lines 2x − 3y+1 = 0 and 5x + y − 3 = 0 Solution
(7) Find the separate equation of the following pair of straight lines
(i) 3x^{2} + 2xy − y^{2} = 0 Solution
(ii) 6(x − 1)^{2} + 5(x − 1)(y − 2) − 4(y − 2)^{2} = 0 Solution
(iii) 2x^{2} − xy − 3y^{2} − 6x + 19y − 20 = 0 Solution
(8) The slope of one of the straight lines ax^{2} + 2hxy + by^{2} = 0 is twice that of the other, show that 8h^{2} = 9ab. Solution
(9) The slope of one of the straight lines ax^{2} + 2hxy + by^{2} = 0 is three times the other, show that 3h^{2} = 4ab. Solution
(10) A ΔOPQ is formed by the pair of straight lines x^{2} −4xy +y^{2}= 0 and the line PQ. The equation of PQ is x + y − 2 = 0. Find the equation of the median of the triangle ΔOPQ drawn from the origin O. Solution
(11) Find p and q, if the following equation represents a pair of perpendicular lines
6x^{2} + 5xy − py^{2} + 7x + qy − 5 = 0 Solution
(12) Find the value of k, if the following equation represents a pair of straight lines. Further, find whether these lines are parallel or intersecting, 12x^{2} + 7xy − 12y^{2} − x + 7y + k = 0 Solution
(13) For what value of k does the equation 12x^{2}+2kxy+2y^{2}+11x−5y+2 = 0 represent two straight lines.
(14) Show that the equation 9x^{2 }− 24xy + 16y^{2 }− 12x + 16y − 12 = 0 represents a pair of parallel lines. Find the distance between them. Solution
(15) Show that the equation 4x^{2} + 4xy + y^{2} − 6x − 3y −4 = 0 represents a pair of parallel lines. Find the distance between them. Solution
(16) Prove that one of the straight lines given by ax^{2} + 2hxy + by^{2} = 0 will bisect the angle between the co-ordinate axes if (a + b)^{2} = 4h^{2 }Solution
(17) If the pair of straight lines x^{2} − 2kxy − y^{2} = 0 bisect the angle between the pair of straight lines x^{2} − 2lxy − y^{2} = 0, Show that the later pair also bisects the angle between the former. Solution
(18) Prove that the straight lines joining the origin to the points of intersection of 3x^{2} + 5xy − 3y^{2} + 2x + 3y = 0 and 3x − 2y − 1 = 0 are at right angles. Solution
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