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Question
Show that the following equation represents a pair of line. Find the acute angle between them:
9x2 - 6xy + y2 + 18x - 6y + 8 = 0
Solution
Comparing this equation with
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 9, h = -3, b = 1, g = 9, f = - 3 and c = 8
∴ D = `|("a","h","g"),("h","b","f"),("g","f","c")|`
`= |(9,-3,9),(-3,1,-3),(9,-3,8)|`
= 9(8 - 9) + 3(- 24 + 27) + 9(9 - 9)
= 9(-1) + 3(3) + 9(0)
= - 9 + 9 + 0 = 0
and h2 - ab = (- 3)2 - 9(1) = 9 - 9 = 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((-3)^2 - 9(1)))/(10)|`
`= |(2sqrt(9 - 9))/10| = 0`
∴ tan θ = tan 0°
∴ θ = 0°.
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