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Find ‘k’, if the equation kxy + 10x + 6y + 4 = 0 represents a pair of straight lines.

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#### Solution

Given equation is kxy + 10x + 6y + 4 = 0

Comparing with ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0, we get

a = 0, h = k/2 , b = 0, g = 5, f = 3, c = 4

Since the given equation represents a pair of lines.

abc + 2fgh - af^{2} - bg^{2} -ch^{2} = 0

(0)(0)(4)+2(3)(5)(k/2)-(0)(3)^2-(0)(5)^2-4(k/2)^2=0

15k - k^{2} = 0

- k^{2} + 15k = 0

- k(k - 15) = 0

k = 0 or k = 15

If k = 0, then the equation becomes

10x + 6y + 4 = 0 which does not represents a pair of lines.

k ≠ 0

Hence, k = 15.

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