Advertisements
Advertisements
प्रश्न
Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`
Advertisements
उत्तर
Let l1 and l2 be the two lines. Slopes of l1 is 1 + `sqrt3` and that of l2 is 1 - `sqrt3`
Therefore the equation of a line (l1) passing through the origin and having slope is
y = `(1 + sqrt3)"x"`
∴ `(1 + sqrt3)"x" - "y" = 0` ...(1)
Similarly, the equation of the line (l2) passing through the origin and having slope is
y = `(1 - sqrt3)"x"`
∴ `(1 - sqrt3)"x" - "y" = 0` ...(2)
From (1) and (2) the required combined equation is
`[(1 + sqrt3)"x" - "y"][(1 - sqrt3)"x" - "y"] = 0`
∴ `(1 + sqrt3)"x"[(1 - sqrt3)"x" - "y"] - "y"[(1 - sqrt3)"x" - "y"] = 0`
∴ `(1 - sqrt3)(1 + sqrt3)"x"^2 - (1 + sqrt3)"xy" - (1 - sqrt3)"xy" + "y"^2 = 0`
∴ `((1)^2 - (sqrt3)^2)"x"^2 - [(1 + sqrt3) + (1 - sqrt 3)]"xy" + "y"^2 = 0`
∴ `(1 - 3)"x"^2 - 2"xy" + "y"^2 = 0`
∴ `- 2"x"^2 - "2 xy" + "y"^2 = 0`
∴ 2x2 + 2xy - y2 = 0
This is the required combined equation.
APPEARS IN
संबंधित प्रश्न
Find the combined equation of the following pair of lines:
2x + y = 0 and 3x − y = 0
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 point (2, 3) and parallel to the coordinate axes.
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:
5x2 – 9y2 = 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:
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
If the slope of one of the two lines given by `"x"^2/"a" + "2xy"/"h" + "y"^2/"b" = 0` is twice that of the other, then ab : h2 = ______.
The joint equation of the lines through the origin and perpendicular to the pair of lines 3x2 + 4xy – 5y2 = 0 is _______.
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 distance between lines (x - 2y)2 + k(x - 2y) = 0 is 3 units, then k = ______.
Find the joint equation of the line passing through (1, 2) and parallel to the coordinate axes
Find the joint equation of the line which are at a distance of 9 units from the Y-axis.
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:
4x2 + 4xy + y2 = 0
Show that the following equations represent a pair of line:
`"x"^2 - 2sqrt3"xy" - "y"^2 = 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:
3x2 - y2 = 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 one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.
Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.
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:
(x - 3)2 + (x - 3)(y - 4) - 2(y - 4)2 = 0
Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.
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.
Find k if the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.
The combined equation of the two lines passing through the origin, each making angle 45° and 135° with the positive X-axis is ______
The joint equation of pair of lines having slopes `1+sqrt2` and `1-sqrt2` 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 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
