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Question
If the line x + 2 = 0 coincides with one of the lines represented by the equation x2 + 2xy + 4y + k = 0, then prove that k = - 4.
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Solution
One of the lines represented by x2 + 2xy + 4y + k = 0 is x + 2 = 0. .....(1)
Let the other line represented by (1) be ax + by + c = 0
∴ their combined equation is (x + 2)(ax + by + c) = 0
∴ ax2 + bxy + cx + 2ax + 2by + 2c = 0
∴ ax2 + bxy + (2a + c)x + 2by + 2c = 0 ...(2)
As the equations (1) and (2) are the combined equations of the same two lines, they are identical.
∴ by comparing their corresponding coefficients, we get,
`"a"/1 = "b"/2 = "2b"/4 = "2c"/"k"` and 2a + c = 0
∴ a = `"2c"/"k"` and c = - 2a
∴ a = `(2(- 2"a"))/"k"`
∴ `1 = (- 4)/"k"`
∴ k = - 4
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