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Find k, the slopes of the lines represented by 3x2 + kxy - y2 = 0 differ by 4. - Mathematics and Statistics

Sum

Find k, the slopes of the lines represented by 3x2 + kxy - y2 = 0 differ by 4.

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Solution

Comparing the equation 3x2 + kxy - y2 = 0 with ax2 + 2hxy - by2 = 0, we get, a = 3, 2h = k, b = -1.

Let m1 and m2 be the slopes of the lines represented by 3x2 + kxy - y2 = 0 

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

and m1m2 = `"a"/"b" = 3/(-1) = -3`

∴ (m1 - m2)2 = (m1 + m2)2 - 4m1m

= k2 - 4 (- 3)

= k2 + 12        ....(1)

But |m1 - m2| = 4

∴ (m1 - m2)2 = 16    ....(2)

∴ from (1) and (2), k2 + 12 = 16

∴ k2 = 4 

∴ k = ± 2  

Concept: Homogeneous Equation of Degree Two
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