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Question
Show that the following equation represents a pair of line. Find the acute angle between them:
2x2 + xy - y2 + x + 4y - 3 = 0
Solution
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)
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