Advertisements
Advertisements
प्रश्न
If θ is the acute angle between the lines represented by ax2 + 2hxy + by2 = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`
उत्तर
Proof: Let m1 and m2 be slopes of lines represented by the equation ax2 + 2hxy + by2 = 0
∴ m1 + m2 = `- (2h)/b` and m1m2 = `a/b`
∴ (m1 – m2)2 = (m1 + m2)2 – 4m1m2
= `(-(2h)/b)^2 - 4(a/b)`
= `(4h^2)/b^2 - (4ab)/b^2`
= `(4h^2 - 4ab)/b^2`
= `(4(h^2 - ab))/b^2`
∴ m1 – m2 = `± (2sqrt(h^2 - ab))/b`
As θ is the acute angle between the lines,
tan θ = `|(m_1 - m_2)/(1 + m_1m_2)|`
= `|(± (2sqrt(h^2 - ab))/b)/(1 + a/b)|`
= `|(2sqrt(h^2 - ab))/(a + b)|`
APPEARS IN
संबंधित प्रश्न
. Show that the lines represented by 3x2 - 4xy - 3y2 = 0 are perpendicular to each other.
Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0`
Find the measure of the acute angle between the line represented by:
2x2 + 7xy + 3y2 = 0
Find the measure of the acute angle between the line represented by:
4x2 + 5xy + y2 = 0
Find the measure of the acute angle between the line represented by:
(a2 - 3b2)x2 + 8abxy + (b2 - 3a2)y2 = 0
Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0
If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 = 0, then show that 100 (h2 - ab) = (a + b)2.
Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.
Find the joint equation of the pair of lines which bisect angles between the lines given by x2 + 3xy + 2y2 = 0
Show that the lines x2 − 4xy + y2 = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle.
Show that the line 3x + 4y + 5 = 0 and the lines (3x + 4y)2 - 3(4x - 3y)2 = 0 form the sides of an equilateral triangle.
Show that the lines x2 - 4xy + y2 = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.
Prove that the product of length of perpendiculars drawn from P(x1, y1) to the lines represented by ax2 + 2hxy + by2 = 0 is `|("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|`
Show that the difference between the slopes of the lines given by (tan2θ + cos2θ)x2 - 2xy tan θ + (sin2θ)y2 = 0 is two.
The acute angle between the lines represented by x2 + xy = 0 is ______.
Find the measure of the acute angle between the lines given by x2 − 4xy + y2 = 0
If θ is the acute angle between the lines given by ax2 + 2hxy + by2 = 0 then prove that tan θ = `|(2sqrt("h"^2) - "ab")/("a" + "b")|`. Hence find acute angle between the lines 2x2 + 7xy + 3y2 = 0
If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 − 5xy + 3y2 = 0, then show that 100(h2 − ab) = (a + b)2
The angle between the pair of straight lines 2x2 - 6xy + y2 = 0 is tan-1 (p), where p = ______
The angle between lines `(x - 2)/2 = (y - 3)/(- 2) = (z - 5)/1` and `(x - 2)/1 = (y - 3)/2 = (z - 5)/2` is ______.
If 4ab = 3h2, then the ratio of slopes of the lines represented by the equation ax2 +2hxy + by2 = 0 will be ______
The acute angle between lines x - 3 = 0 and x + y = 19 is ______.
Which of the following pair of straight lines intersect at right angles?
If slopes of lines represented by kx2 + 5xy + y2 = 0 differ by 1, then k = ______.
The joint equation of the angle bisectors of the angles between the lines 4x2 – 16xy + 7y2 = 0 is ______.
If the lines represented by 5x2 – 3xy + ky2 = 0 are perpendicular to each other, find the value of k.
