Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let m1 and m2 be the slopes of the lines represented by ax2 + 2hxy + by2 = 0.
∴ m1 + m2 = `(- 2"h")/"b"` and m1m2 = `"a"/"b"` ....(1)
Now, the required lines are perpendicular to these lines.
∴ their slopes are `- 1/"m"_1` and `- 1/"m"_2`
Since these lines are passing through the origin, their separate equations are
y = `- 1/"m"_1`x and y = `- 1/"m"_2`x
i.e. m1y = - x and m2y = - x
i.e. x + m1y = 0 and x + m2y = 0
∴ their combined equation is
(x + m1y)(x + m2y) = 0
∴ x2 + (m1 + m2)xy + m1m2y2 = 0
∴ `"x"^2 (-"2h")/"b" "x" + "a"/"b" "y"^2 = 0` ...[By(1)]
∴ bx2 - 2hxy + ay2 = 0
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 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:
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
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
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 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 ______.
The area of triangle formed by the lines x2 + 4xy + y2 = 0 and x - y - 4 = 0 is ______.
Choose correct alternatives:
The combined equation of the coordinate axes 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 = 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
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:
x2 - y2 = 0
Show that the following equations represent a pair of line:
`"x"^2 - 2sqrt3"xy" - "y"^2 = 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 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 the combined equation of bisectors of angles between the lines represented by 5x2 + 6xy - y2 = 0.
Find a if the sum of the slopes of lines represented by ax2 + 8xy + 5y2 = 0 is twice their product.
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
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.
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.
If equation ax2 - y2 + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.
Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`
The combined equation of the lines through origin and perpendicular to the pair of lines 3x2 + 4xy − 5y2 = 0 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 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 ______.
The joint equation of pair of lines having slopes `1+sqrt2` and `1-sqrt2` and passing through the origin is ______.
Write the separate equations of lines represented by the equation 5x2 – 9y2 = 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.
