Advertisements
Advertisements
Question
Find p and q, if the equation 2x2 + 8xy + py2 + qx + 2y - 15 = 0 represents a pair of parallel lines.
Solution
The given equation is 2x2 + 8xy + py2 + qx + 2y - 15 = 0
Comparing it with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 2, h = 4, b = p, g = `"q"/2`, f = 1, c = - 15
Since the lines are parallel, h2 = ab
∴ (4)2 = 2p
∴ p = 8
Since the given equation represents a pair of lines
D = `|("a","h","g"),("h","b","f"),("g","f","c")| = 0` where b = p = 8
i.e. `|(2,4,"q"/2),(4,8,1),("q"/2,1,-15)| = 0`
i.e. `2(- 120 - 1) - 4 (- 60 - "q"/2) + "q"/2(4 - "4q") = 0`
i.e. - 242 + 240 + 2q + 2q - 2q2 = 0
i.e. - 2q2 + 4q - 2 = 0
i.e. q2 - 2q + 1 = 0
i.e. (q - 1)2 = 0
∴ q - 1 = 0
∴ q = 1
Hence, p = 8 and q = 1
APPEARS IN
RELATED QUESTIONS
Find p and q if the equation px2 – 8xy + 3y2 + 14x + 2y + q = 0 represents a pair of prependicular lines.
Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x2 + 3xy - 9y2 = 0.
Find the joint equation of the pair of the line through the point (2, -3) and parallel to the lines represented by x2 + xy - y2 = 0.
Show that the equation x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 does not represent a pair of lines.
Show that the equation 2x2 + xy - y2 + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.
Find the separate equation of the line represented by the following equation:
(x - 2)2 - 3(x - 2)(y + 1) + 2(y + 1)2 = 0
Find the value of k, if the following equations represent a pair of line:
3x2 + 10xy + 3y2 + 16y + k = 0
Find the value of k, if the following equations represent a pair of line:
kxy + 10x + 6y + 4 = 0
Find the value of k, if the following equations represent a pair of line:
x2 + 3xy + 2y2 + x - y + k = 0
Equations of pairs of opposite sides of a parallelogram are x2 - 7x + 6 = 0 and y2 − 14y + 40 = 0. Find the joint equation of its diagonals.
Find the separate equation of the line represented by the following equation:
10(x + 1)2 + (x + 1)(y - 2) - 3(y - 2)2 = 0
ΔOAB is formed by the lines x2 - 4xy + y2 = 0 and the line AB. The equation of line AB is 2x + 3y - 1 = 0. Find the equation of the median of the triangle drawn from O.
Find the coordinates of the points of intersection of the lines represented by x2 − y2 − 2x + 1 = 0
The point of intersection of lines given by the equation 2x2 + 4xy - 2y2 + 4x + 8y + 1 = 0 is ______
If (a + 3b)(3a + b) = 4h2, then the angle between the lines represented by ax2 + 2hxy + by2 = 0 is ______
9x2 + hxy + y2 = 0 represents pair of parallel straight lines, if h is ______.
Let L be the line joining the origin to the point of intersection of the lines represented by 2x2 – 3xy – 2y2 + 10x + 5y = 0. lf L is perpendicular to the line kx + y + 3 = 0, then k = ______.
Equation of line passing through the points (0, 0, 0) and (2, 1, –3) is ______.
ΔOAB is formed by lines x2 – 4xy + y2 = 0 and the line x + y – 2 = 0. Find the equation of the median of the triangle drawn from O.
Find the coordinates of the point of intersection of the pair of lines 6x2 + 5xy – 4y2 + 7x + 13y – 3 = 0.
Find the point of intersection of the lines given by x2 + 3xy + 2y2 + x – y – 6 = 0
Find the values of p and q if the equation px2 – 6xy + y2 + 18x – qy + 8 = 0 represents a pair of parallel lines.
