Advertisements
Advertisements
प्रश्न
Show that the following equation represents a pair of line. Find the acute angle between them:
2x2 + xy - y2 + x + 4y - 3 = 0
Advertisements
उत्तर
Comparing this equation with
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 2, h = `1/2`, b = - 1, g = `1/2`, f = 2 and c = -3
Now, abc + 2fgh - af2 - bg2 - ch2 = 0
`=2(-1)(-3)+2(2)(1/2)(1/2)-2(2)^2-(-1)(1/2)^2-(-3)(1/2)^2`
`=6+1-8+1/4+3/4`
`=7-8+1`
`=8-8=0`
And, `h^2-ab=(1/2)^2-(2)(-1)`
`=1/4+2=9/4>0`
∴ the given equation represents a pair of lines.
Let θ be the acute angle between the lines.
∴ tan θ = `|(2sqrt("h"^2 - "ab"))/("a + b")|`
`= |(2sqrt((9/4)))/(2 - 1)|`
`= |2 xx3/2| = 3`
∴ tan θ = tan 3
∴ θ = tan-1 (3)
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 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 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:
If two lines ax2 + 2hxy + by2 = 0 make angles α and β with X-axis, then tan (α + β) = _____.
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 ______.
Choose correct alternatives:
If h2 = ab, then slopes of lines ax2 + 2hxy + by2 = 0 are in the ratio
Choose correct alternatives:
If slope of one of the lines ax2 + 2hxy + by2 = 0 is 5 times the slope of the other, then 5h2 = ______
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 having slopes 2 and 3.
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 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:
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 separate equation of the line represented by the following equation:
2x2 + 2xy - y2 = 0
Find k, if the sum of the slopes of the lines given by 3x2 + kxy - y2 = 0 is zero.
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 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.
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 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.
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`
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 the lines through the origin which forms two of the sides of the equilateral triangle having x = 2 as the third side is ______
The joint equation of pair of lines having slopes 2 and 5 and passing through the origin is ______.
The combined equation of the lines which pass through the origin and each of which makes an angle of 30° with the line 3x + 2y – 11 = 0 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 ______.
Write the joint equation of co-ordinate axes.
Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by 3x2 + 2xy – y2 = 0.
Find the combined equation of y-axis and the line through the origin having slope 3.
