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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)
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)
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