Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let OA and OB be the required lines.
Let OA (or OB) has slope m.
∴ its equation is y = mx ...(1)
It makes an angle α with x + y = 0 whose slope is - 1.
∴ tan α = `|("m" + 1)/(1 + "m"(- 1))|`
Squaring both sides, we get,
`"tan"^2alpha = ("m" + 1)^2/(1 - "m")^2`
∴ tan2α (1 - 2m + m2) = m2 + 2m + 1
∴ tan2α - 2mtan2α + m2tan2α = m2 + 2m + 1
∴ (tan2α - 1)m2 - 2(1 + tan2α)m + (tan2α - 1) = 0
∴ `"m"^2 - 2((1 + "tan"^2alpha)/("tan"^2alpha - 1))"m" + 1 = 0`
∴ `"m"^2 + 2((1 + "tan"^2alpha)/(1 - "tan"^2alpha)) "m" + 1 = 0`
∴ `"m"^2 + 2(sec 2 alpha)"m" + 1 = 0` ...`[because "cos 2"alpha = (1 - "tan"^2 alpha)/(1 + "tan"^2alpha)]`
∴ `"y"^2/"x"^2 + 2("sec"2alpha)"y"/"x" + 1 = 0`
∴ `"y"^2 2"xy" "sec" 2 alpha + "x"^2 = 0` ...[By (1)]
∴ `"y"^2 + 2"xy" "sec 2" alpha + "x"^2 = 0`
∴ x2 + 2(sec 2α)xy + y2 = 0 is the required 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 lines passing through point (2, 3) and parallel to the coordinate axes.
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:
`3"x"^2 - 2sqrt3"xy" - 3"y"^2 = 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 = ______.
Choose correct alternatives:
The combined equation of the coordinate axes is
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 (1, 2) and parallel to the coordinate axes
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 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
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 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 the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.
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.
Show that the following equation represents a pair of line. Find the acute angle between them:
2x2 + xy - y2 + x + 4y - 3 = 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.
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 equation ax2 - y2 + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.
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 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 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 ______.
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
