Advertisements
Advertisements
Question
Show that the equation 2x2 − xy − 3y2 − 6x + 19y − 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is tan−1(5)
Advertisements
Solution
TThe equation of the given pair of lines is
2x2 – xy – 3y2 – 6x + 19y – 20 = 0 .......(1)
Compare this equation with the equation
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 .......(2)
a = 2
2h = – 1
b = – 3
2g = – 6
2f = 19
c = – 20
h2 – ab = `(- 1/2)^2 - 2(2)( - 3)`
= `1/4 + 6 ≠ 0`
∴ The given line (1) is not parallel.
∴ They are intersecting lines.
Let θ be the angle between the lines.
tan θ = `(2sqrt("h"^2 - "ab"))/("a" + "b")`
tan θ = `root(2)(( - 1/2)^2 - (2)( - 3))/(2 + ( - 3))`
= `root(2)(1/4 + 6)/(- 1)`
= `- 2sqrt((1 + 24)/4`
tan θ = ` - 2/2 sqrt(25)`
= – 5
Taking the acute angle θ = tan−1(5)
APPEARS IN
RELATED QUESTIONS
Show that the equation 12x2 – 10xy + 2y2 + 14x – 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.
Find the angle between the pair of straight lines 3x2 – 5xy – 2y2 + 17x + y + 10 = 0.
If m1 and m2 are the slopes of the pair of lines given by ax2 + 2hxy + by2 = 0, then the value of m1 + m2 is:
Show that 4x2 + 4xy + y2 − 6x − 3y − 4 = 0 represents a pair of parallel lines
Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x2 – 2xy sec 2α + y2 = 0
Find the separate equation of the following pair of straight lines
3x2 + 2xy – y2 = 0
Find the separate equation of the following pair of straight lines
6(x – 1)2 + 5(x – 1)(y – 2) – 4(y – 3)2 = 0
The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is twice that of the other, show that 8h2 = 9ab
Find p and q, if the following equation represents a pair of perpendicular lines
6x2 + 5xy – py2 + 7x + qy – 5 = 0
Show that the equation 4x2 + 4xy + y2 – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them
If the pair of straight lines x2 – 2kxy – y2 = 0 bisect the angle between the pair of straight lines x2 – 2lxy – y2 = 0, Show that the later pair also bisects the angle between the former
Choose the correct alternative:
The coordinates of the four vertices of a quadrilateral are (−2, 4), (−1, 2), (1, 2) and (2, 4) taken in order. The equation of the line passing through the vertex (−1, 2) and dividing the quadrilateral in the equal areas is
Choose the correct alternative:
If the equation of the base opposite to the vertex (2, 3) of an equilateral triangle is x + y = 2, then the length of a side is
Choose the correct alternative:
One of the equation of the lines given by x2 + 2xy cot θ – y2 = 0 is
