Advertisements
Advertisements
Question
Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.
Advertisements
Solution
Let OA and OB be the lines through the origin making an angle of 60° with the line x = 3.
∴ OA and OB make an angle of 30° and 150° with the positive direction of X-axis.
∴ Slope of OA = tan 30° = `1/sqrt3`
∴ Equation of the line OA is y = `1/sqrt3x`
∴ `sqrt3y` = x
∴ `x - sqrt3"y" = 0`
Slope of OB = tan 150°
= tan (180° – 30°)
= – tan 30°
= `- 1/sqrt3`
∴ Equation of the line OB is y = `-1/sqrt3x`
∴ `sqrt3y` = – x
∴ x + `sqrt3y` = 0
∴ Required combined equation of the lines is
`(x - sqrt3y)(x + sqrt3y)` = 0
i.e. x2 – 3y2 = 0
APPEARS IN
RELATED QUESTIONS
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:
`3"x"^2 - 2sqrt3"xy" - 3"y"^2 = 0`
Find the separate equation of the line represented by the following equation:
x2 + 2(cosec α)xy + y2 = 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 the following equation:
xy + y2 = 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
Choose correct alternatives:
Auxiliary equation of 2x2 + 3xy - 9y2 = 0 is
Choose correct alternatives:
If two lines ax2 + 2hxy + by2 = 0 make angles α and β with X-axis, then tan (α + β) = _____.
Choose correct alternatives:
The combined equation of the coordinate axes is
Find the joint equation of the line:
x - y = 0 and x + y = 0
Find the joint equation of the line:
x + y − 3 = 0 and 2x + y − 1 = 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 (1, 2) and parallel to the coordinate axes
Find the joint equation of the line passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18
Find the joint equation of the line passing through (-1, 2) and perpendicular to the lines x + 2y + 3 = 0 and 3x - 4y - 5 = 0
Show that the following equations represents a pair of line:
x2 + 2xy - y2 = 0
Find the separate equation of the line represented by the following equation:
x2 - 4y2 = 0
Find the separate equation of the line represented by the following equation:
2x2 + 2xy - y2 = 0
Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by
x2 + 4xy - 5y2 = 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 k, if one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.
If the line 4x - 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0, then show that 25a + 40h + 16b = 0
Show that the following equation represents a pair of line. Find the acute angle between them:
2x2 + xy - y2 + x + 4y - 3 = 0
Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.
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.
Find k if the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.
Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`
The combined equation of the two lines passing through the origin, each making angle 45° and 135° with the positive X-axis is ______
Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0
The joint equation of the lines through the origin which forms two of the sides of the equilateral triangle having x = 2 as the third side 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 ______.
The joint equation of pair of lines having slopes 2 and 5 and passing through the origin is ______.
The line 5x + y – 1 = 0 coincides with one of the lines given by 5x2 + xy – kx – 2y + 2 = 0 then the value of k is ______.
Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by 3x2 + 2xy – y2 = 0.
Combined equation of the lines bisecting the angles between the coordinate axes, is ______.
Find the combined equation of y-axis and the line through the origin having slope 3.
The distance between the lines represented by the equation 4x² + 4xy + y² − 6x − 3y − 4 = 0 is
