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Question
If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 = 0, then show that 100 (h2 - ab) = (a + b)2.
Solution
The acute angle θ between the lines ax2 + 2hxy + by2 = 0 is given by
tan θ = `|(2sqrt("h"^2 - "ab"))/("a + b")|` ....(1)
Comparing the equation 2x2 - 5xy + 3y2 = 0 with ax2 + 2hxy + by2 = 0, we get.
a = 2, 2h = -5, i.e. h = `- 5/2` and b = 3
Let α be the acute angle between the lines 2x2 - 5xy + 3y2 = 0
∴ tan α = `|(2sqrt("h"^2 - "ab"))/("a + b")|`
`= |(2 sqrt((5/2)^2 - 2(3)))/(2 + 3)|`
`= |((2 sqrt25/4 - 6))/5|`
`= |(2 xx 1/2)/5|`
∴ tan α = `1/5` ....(2)
If θ = α, then tan θ = tan α
`therefore |(2sqrt("h"^2 - "ab"))/("a + b")| = 1/5` ....[By (1) and (2)]
∴ `(4("h"^2 - "ab"))/("a + b")^2 = 1/25`
∴ 100 (h2 - ab) = (a + b)2
This is the required condition.
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