Advertisements
Advertisements
Question
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.
Solution
The combined equation of the given lines is
x2 + xy - y2 = 0 ....(1)
with ax2 + 2hxy + by2 = 0, we get,
a = 1, 2h = 1, b = - 1
Let m1 and m2 be the slopes of the lines represented by (1).
Then m1 + m2 = `- "2h"/"b" = (-1)/-1 = 1` and m1m2 = `"a"/"b" = 1/-1 = - 1` .....(2)
The slopes of the lines parallel to these lines are m1 and m2 .
∴ the equations of the lines with these slopes and through the point (2, - 3) are
y + 3 = m1(x - 2) and y + 3 = m2 (x - 2)
i.e. m1(x - 2) - (y + 3) = 0 and m2(x - 2) - (y + 3) = 0
∴ the joint equation of these lines is
[m1 (x - 2) - (y + 3)][m2(x - 2)- (y + 3)] = 0
∴ m1m2 (x - 2)2 - m1 (x - 2)(y + 3) - m2(x - 2)(y+3) + (y + 3)2 = 0
∴ m1m2 (x - 2)2 - (m1 + m2)(x - 2)(y + 3) + (y + 3)3 = 0
∴ `- ("x" - 2)^2 - ("x" - 2)("y" + 3) + ("y" + 3)^2 = 0` ....[By (2)]
∴ `("x" - 2)^2 + ("x" - 2)("y" + 3) - ("y" + 3)^2 = 0`
∴ `("x"^2 - "4x" + 4) + ("xy" + "3x" - "2y" - 6) - ("y"^2 + "6y" + 9) = 0`
∴ `"x"^2 - "4x" + 4 + "xy" + "3x" - "2y" - 6 - "y"^2 - "6y" - 9 = 0`
∴ `"x"^2 + "xy" - "y"^2 - "x" - "8y" - 11 = 0`
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.
Show that the equation x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 does not represent a pair of lines.
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
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.
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 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 ______.
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.
