Advertisements
Advertisements
Question
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
Solution
Let m1 and m2 be the slopes of the lines represented by the equation ax2 + 2hxy + by2 = 0.
∴ m1 + m2 = `(-2"h")/"b"` and m1m2 = `"a"/"b"`
∴ (m1 − m2)2 = (m1 + m2)2 – 4m1m2
= `((-2"h")/"b")^2 - 4("a"/"b")`
= `(4"h"^2)/"b"^2 - (4"ab")/"b"^2`
= `(4"h"^2 - 4"ab")/"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"_1"m"_2)|`
= `|(± (2sqrt("h"^2 - "ab"))/"b")/(1 + "a"/"b")|`
= `|(2sqrt("h"^2 - "ab"))/("a" + "b")|` .......(i)
Comparing the equation 2x2 + 7xy + 3y2 = 0 with ax2 + 2hxy + by2 = 0,
We get a = 2, h = `7/2`, b = 3
Substituting in equation (i), we get
tan θ = `|(2sqrt((7/2)^2 - 2(3)))/(2 + 3)|`
= `|(2sqrt((49/4) - 6))/5|`
= `|(2sqrt((49 - 24)/4))/5|`
= `|(2* 5/2)/5|`
= 1
∴ θ = tan–1(1) = 45°
RELATED QUESTIONS
Show that the lines represented by x2 + 6xy + 9y2 = 0 are coincident.
Find the value of k if lines represented by kx2 + 4xy – 4y2 = 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:
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
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
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is three times the other, prove that 3h2 = 4ab.
Show that the lines x2 - 4xy + y2 = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is square of the slope of the other line, show that a2b + ab2 + 8h3 = 6abh.
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 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 the line `x/(3) = y/(4)` = z is perpendicular to the line `(x - 1)/k = (y + 2)/(3) = (z - 3)/(k - 1)`, then the value of k is ______.
The acute angle between the curve x = 2y2 and y = 2x2 at `(1/2, 1/2)` is ______.
If slopes of lines represented by kx2 + 5xy + y2 = 0 differ by 1, then k = ______.
Find the combined equation of the pair of lines through the origin and making an angle of 30° with the line 2x – y = 5
If θ is the acute angle between the lines given by 3x2 – 4xy + by2 = 0 and tan θ = `1/2`, find b.
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.
