PAIR OF STRAIGHT LINES PROBLEMS WITH SOLUTION

(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 4x2 + 4xy + y2 − 6x − 3y − 4 = 0 represents a pair of parallel lines.      Solution

(3)  Show that 2x2 + 3xy − 2y2 + 3x + y + 1 = 0 represents a pair of perpendicular lines.     Solution

(4)  Show that the equation 2x2 −xy−3y2 −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 x2 − 2xy sec 2α + y2 = 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) 3x2 + 2xy − y2 = 0               Solution

(ii)  6(x − 1)2 + 5(x − 1)(y − 2) − 4(y − 2)2 = 0     Solution

(iii) 2x2 − xy − 3y2 − 6x + 19y − 20 = 0       Solution

(8)  The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is twice that of the other, show that 8h2 = 9ab.   Solution

(9)  The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is three times the other, show that 3h2 = 4ab.   Solution    

(10) A ΔOPQ is formed by the pair of straight lines x2 −4xy +y2= 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

6x2 + 5xy − py2 + 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, 12x2 + 7xy − 12y2 − x + 7y + k = 0           Solution

(13)  For what value of k does the equation 12x2+2kxy+2y2+11x−5y+2 = 0 represent two straight lines.

Solution

(14)  Show that the equation 9x− 24xy + 16y− 12x + 16y − 12 = 0 represents a pair of parallel lines. Find the distance between them.      Solution

(15)  Show that the equation 4x2 + 4xy + y2 − 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 ax2 + 2hxy + by2 = 0 will bisect the angle between the co-ordinate axes if (a + b)2 = 4h2            Solution

(17)  If the pair of straight lines x2 − 2kxy − y2 = 0 bisect the angle between the pair of straight lines x2 − 2lxy − y2 = 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 3x2 + 5xy − 3y2 + 2x + 3y = 0 and 3x − 2y − 1 = 0 are at right angles.        Solution

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