Advertisements
Advertisements
प्रश्न
Show that the following equations represent a pair of line:
x2 + 7xy - 2y2 = 0
Advertisements
उत्तर
Comparing the equation x2 + 7xy - 2y2 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = 1, 2h = 7 i,e, h = `7/2`, and b = - 2
∴ h2 - ab = `(7/2)^2` - 1 (- 2)
`= 49/4 + 2`
`= 57/4` i.e. 14.25 = 14 > 0
Since the equation x2 + 7xy - 2y2 = 0 is a homogeneous equation of second degree and h2 - ab > 0, the given equation represents a pair of lines which are real and distinct.
APPEARS IN
संबंधित प्रश्न
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 lines:
Passing through (2, 3) and perpendicular to the lines 3x + 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:
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:
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) = _____.
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:
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 h2 = ab, then slopes of lines ax2 + 2hxy + by2 = 0 are in the ratio
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 the origin and having inclinations 60° and 120°.
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:
6x2 - 5xy - 6y2 = 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 + 4xy - 5y2 = 0
Find k, if the sum of the slopes of the lines given by 3x2 + kxy - y2 = 0 is zero.
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 joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.
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:
(x - 3)2 + (x - 3)(y - 4) - 2(y - 4)2 = 0
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.
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 combined equation of the lines through origin and perpendicular to the pair of lines 3x2 + 4xy − 5y2 = 0 is ______
The joint equation of pair of straight lines passing through origin and having slopes `(1 + sqrt2) and (1/(1 + sqrt2))` is ______.
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 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 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 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.
