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Question
Show that the following equation represents a pair of line. Find the acute angle between them:
(x - 3)2 + (x - 3)(y - 4) - 2(y - 4)2 = 0
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Solution
Put x - 3 = X and y - 4 = Y in the given equation, we get,
X2 + XY - 2Y2= 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we get,
a = 1, h = 1/2, b = - 2
This is the homogeneous equation of second degree
and h2 - ab = `(1/2)^2 - 1(- 2)`
`= 1/4 + 2 = 9/4 > 0`
Hence, it represents a pair of lines passing through the new origin (3, 4).
Let θ be the acute angle between the lines.
∴ tan θ = `|(2 sqrt("h"^2 - "ab"))/("a + b")|`
here a = 1, 2h = 1, i.e. h = `1/2` and b = - 2
∴ tan θ = `|(2sqrt((1/2)^2 - 1(-2)))/(1 - 2)|`
`= |(2(sqrt(1/4 + 2)))/-1|`
`= |(2 xx 3/2)/-1|`
∴ tan θ = 3
∴ θ = tan-1(3)
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