Advertisements
Advertisements
Question
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.
Solution
The combined equation of the given lines is
2x2 + 3xy - 9y2 = 0
i.e. 2x2 + 6xy - 3xy - 9y2 = 0
i.e. 2x(x + 3y) - 3y(x + 3y) = 0
i.e. (x + 3y)(2x - 3y) = 0
∴ their separate equations are
x + 3y = 0 and 2x - 3y = 0
∴ their slopes are m1 = `(-1)/3` and m2 = `(-2)/-3 = 2/3`
The slopes of the lines parallel to these lines are m1 and m2 i.e. `- 1/3` and `2/3`.
∴ the equations of the lines with these slopes and through the point (2, -1) are
y + 1 = `- 1/3`(x - 2) and y + 1 = `2/3` (x - 2)
i.e. 3y + 3 = - x + 2 and 3y + 3 = 2x - 4
i.e. x + 3y + 1 = 0 and 2x - 3y - 7 = 0
∴ the joint equation of these lines is
(x + 3y + 1)(2x - 3y - 7) = 0
∴ 2x2 - 3xy - 7x + 6xy - 9y2 - 21y + 2x - 3y - 7 = 0
∴ 2x2 + 3xy - 9y2 - 5x - 24y - 7 = 0
APPEARS IN
RELATED QUESTIONS
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 - 3y2 - 6x + 19y - 20 = 0 represents 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 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 p and q, if the equation 2x2 + 8xy + py2 + qx + 2y - 15 = 0 represents a pair of parallel lines.
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
Find the coordinates of the points of intersection of the lines represented by x2 − y2 − 2x + 1 = 0
Find k, if the slopes of lines given by kx2 + 5xy + y2 = 0 differ by 1.
The point of intersection of lines given by the equation 2x2 + 4xy - 2y2 + 4x + 8y + 1 = 0 is ______
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is two times the other, then ______
If (a + 3b)(3a + b) = 4h2, then the angle between the lines represented by ax2 + 2hxy + by2 = 0 is ______
If the equation `lambdax^2 + 2y^2 - 5xy + 5x - 7y + 3 = 0` represents two straight lines, then the value of λ will be ______.
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 = ______.
If ax2 + 2xy – 3y2 + 4x + c = 0 represents a pair of perpendicular lines, then ______.
Equation of line passing through the points (0, 0, 0) and (2, 1, –3) is ______.
The point of intersection of line represented by x2 – y2 + 3y – 2 = 0 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.
