Advertisements
Advertisements
Question
Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.
Advertisements
Solution
Comparing the equation ay2 + bxy + ex + dy = 0 with Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0, we get,
A = 0, H = `"b"/2`, B = a, G = `"e"/2`, F = `"d"/2`, C = 0
The given equation represents a pair of lines,
if `|("A","H","G"),("H","B","F"),("G","F","C")| = 0`
i.e. if `|(0,"b"/2,"e"/2),("b"/2,"a","d"/2),("e"/2,"d"/2, 0)| = 0`
i.e. if `0 - "b"/2(0 - "ed"/4) + "e"/2("bd"/4 - "ae"/2) = 0`
i.e. if `"bed"/8 + "bed"/8 - "ae"^2/4 = 0`
i.e. if bed - ae2 = 0
i.e. if e(bd - ae) = 0
i.e. if e = 0 or bd - ae = 0
i.e. if e = 0 or bd = ae
This is the required condition.
APPEARS IN
RELATED QUESTIONS
Find the combined equation of the following pair of line:
x + 2y - 1 = 0 and x - 3y + 2 = 0
Find the combined equation of the following pair of lines:
Passing through (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0.
Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0
Find the separate equation of the line represented by the following equation:
3y2 + 7xy = 0
Find the separate equation of the line represented by the following equation:
x2 - 4xy = 0
Find the separate equation of the line represented by the following equation:
x2 + 2xy tan α - y2 = 0
Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by following equation:
5x2 - 8xy + 3y2 = 0
Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:
5x2 + 2xy - 3y2 = 0
Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:
3x2 − 4xy = 0
The joint equation of the lines through the origin and perpendicular to the pair of lines 3x2 + 4xy – 5y2 = 0 is _______.
Choose correct alternatives:
If the equation 3x2 – 8xy + qy2 + 2x + 14y + p = 1 represents a pair of perpendicular lines, then the values of p and q are respectively ______.
Choose correct alternatives:
The combined equation of the coordinate axes is
Choose correct alternatives:
If h2 = ab, then slopes of lines ax2 + 2hxy + by2 = 0 are in the ratio
Choose correct alternatives:
If slope of one of the lines ax2 + 2hxy + by2 = 0 is 5 times the slope of the other, then 5h2 = ______
Find the joint equation of the line:
x - y = 0 and x + y = 0
Find the joint equation of the line passing through the origin having slopes 2 and 3.
Find the joint equation of the line passing through (3, 2) and parallel to the lines x = 2 and y = 3.
Find the joint equation of the line passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18
Show that the following equations represents a pair of line:
x2 + 2xy - y2 = 0
Show that the following equations represents a pair of line:
4x2 + 4xy + y2 = 0
Show that the following equations represent a pair of line:
x2 + 7xy - 2y2 = 0
Find the separate equation of the line represented by the following equation:
6x2 - 5xy - 6y2 = 0
Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by
2x2 - 3xy - 9y2 = 0
Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by
x2 + xy - y2 = 0
Find k, if the sum of the slopes of the lines given by x2 + kxy − 3y2 = 0 is equal to their product.
Find k, if one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.
Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0.
Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x2 + 2(sec 2α)xy + y2 = 0
If the lines given by ax2 + 2hxy + by2 = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h2.
Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax2 + 2hxy + by2 = 0 is bx2 - 2hxy + ay2 = 0.
The combined equation of the lines through origin and perpendicular to the pair of lines 3x2 + 4xy − 5y2 = 0 is ______
The joint equation of pair of straight lines passing through origin and having slopes `(1 + sqrt2) and (1/(1 + sqrt2))` is ______.
The joint equation of pair of lines through the origin, each of which makes an angle of 60° with Y-axis, is ______
The equation of line passing through the midpoint of the line joining the points (-1, 3, -2) and (-5, 3, -6) and equally inclined to the axes is ______.
Write the joint equation of co-ordinate axes.
Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by 3x2 + 2xy – y2 = 0.
If `x^2/a + y^2/b + (2xy)/h` = 0 represents a pair of lines and slope of one line is twice the other, then find the value of ab : h2.
Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by 2x2 + 7xy + 3y2 = 0
Find the combined equation of y-axis and the line through the origin having slope 3.
Find k, if one of the lines given by kx2 – 5xy – 3y2 = 0 is perpendicular to the line x – 2y + 3 = 0
The distance between the lines represented by the equation 4x² + 4xy + y² − 6x − 3y − 4 = 0 is
