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Find k, the slope of one of the lines given by kx2 + 4xy - y2 = 0 exceeds the slope of the other by 8. - Mathematics and Statistics

Sum

Find k, the slope of one of the lines given by kx2 + 4xy - y2 = 0 exceeds the slope of the other by 8.

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Solution

Comparing the equation kx2 + 4xy - y2 = 0 with ax2 + 2hxy - by2 = 0, we get, a = k, 2h = 4, b = -1.

Let m1 and m2 be the slopes of the lines represented by kx2 + 4xy - y2 = 0 

∴ m1 + m2 = `(-2"h")/"b" = - 4/(-1) = 4`

and m1m2 = `"a"/"b" = "k"/(-1) = -"k"`

We are given that m2 = m1 + 8

∴ m1 + m1 + 8 = 4 

∴ 2m1 = - 4    ∴ m1 = - 2     ...(1)

Also, m1(m1 + 8) = - k

(-2)(- 2 + 8) = - k      ...[By (1)]

∴ (-2)(6) = - k

∴ - 12 = - k

∴ k = 12

Concept: Homogeneous Equation of Degree Two
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