Advertisements
Advertisements
प्रश्न
Find the combined equation of bisectors of angles between the lines represented by 5x2 + 6xy - y2 = 0.
Advertisements
उत्तर
Comparing the equation 5x2 + 6xy - y2 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = 5, 2h = 6, b = -1
Let m1 and m2 be the slopes of the lines represented by 5x2 + 6xy - y2 = 0.
∴ m1 + m2 = `(-2"h")/"b" = (-6)/-1 = 6` and m1m2 = `"a"/"b" = 5/-1 = - 5` ...(1)
The separate equations of the lines are
y = m1x and y = m2x, where m1 ≠ m2
i.e. m1x - y = 0 and m2x - y = 0.
Let P(x, y) be any point on one of the bisector of the angles between the lines.
∴ the distance of P from the line m1x - y = 0 is equal to the distance of P from the line m2x - y = 0.
∴ `|("m"_1"x" - "y")/(sqrt("m"_1^2 + 1))| = |("m"_2"x" - "y")/(sqrt("m"_2^2 + 1))|`
Squaring both sides, we get,
`("m"_1"x" - "y")^2/("m"_1^2 + 1) = ("m"_2"x" - "y")/("m"_2^2 + 1)`
∴ `("m"_2^2 + 1)("m"_1"x" - "y")^2 = ("m"_1^2 + 1)("m"_2"x" - "y")`
∴ `("m"_2^2 + 1)("m"_1^2"x"^2 - 2"m"_1"xy" + "y"^2) = ("m"_1^2 + 1)("m"_2^2"x"^2 - 2"m"_2"xy" + "y"^2)`
∴ `"m"_1^2"m"_2^2"x"^2 - 2"m"_1"m"_2^2"y"^2"xy" + "m"_2^2"y"^2 + "m"_1^2"x"^2 - 2"m"_1"xy" + "y"^2 = "m"_1^2"m"_2^2"x"^2 - 2"m"_1^2"m"_2"xy" + "m"_1^2"y"^2 + "m"_2^2"x"^2 - 2"m"_2"xy" + "y"^2`
∴ `("m"_1^2 - "m"_2^2)"x"^2 + 2"m"_1"m"_2("m"_1 -"m"_2)"xy" - 2("m"_1 - "m"_2)"xy" - ("m"_1^2 - "m"_2^2)"y"^2 = 0`
Dividing throughout by `"m"_1- "m"_2` (≠ 0), we get,
`("m"_1 + "m"_2)"x"^2 + 2"m"_1"m"_2"xy" - 2"xy" - ("m"_1 + "m"_2)"y"^2 = 0`
∴ 6x2 - 10xy - 2xy - 6y2 = 0 ...[By (1)]
∴ 6x2 - 12xy - 6y2 = 0
∴ x2 - 2xy - y2 = 0
This is the joint equation of the bisectors of the angles between the lines represented by 5x2 + 6xy - y2 = 0.
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 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:
`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 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
Choose correct alternatives:
Auxiliary equation of 2x2 + 3xy - 9y2 = 0 is
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 area of triangle formed by the lines x2 + 4xy + y2 = 0 and x - y - 4 = 0 is ______.
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:
x + y − 3 = 0 and 2x + y − 1 = 0
Find the joint equation of the line passing through the origin and having inclinations 60° and 120°.
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 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 represent a pair of line:
x2 - y2 = 0
Show that the following equations represent a pair of line:
x2 + 7xy - 2y2 = 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:
6x2 - 5xy - 6y2 = 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 the sum of the slopes of the lines given by x2 + kxy − 3y2 = 0 is equal to their product.
Find k, if the slope of one of the lines given by 3x2 - 4xy + ky2 = 0 is 1.
Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0.
Find a if the sum of the slopes of lines represented by ax2 + 8xy + 5y2 = 0 is twice their product.
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 line x + 2 = 0 coincides with one of the lines represented by the equation x2 + 2xy + 4y + k = 0, then prove that k = - 4.
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.
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
Write the separate equations of lines represented by the equation 5x2 – 9y2 = 0
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 k, if one of the lines given by kx2 – 5xy – 3y2 = 0 is perpendicular to the line x – 2y + 3 = 0
