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Question
Find k if the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.
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Solution
3x2 + 4xy + ky2 = 0
∴ divide by x2
`"3x"^2/"x"^2 + "4xy"/"x"^2 + "ky"^2/"x"^2 = 0`
`3 + "4y"/"x" + "ky"^2/"x"^2 = 0` ....(1)
∴ y = mx
∴ `"y"/"x" = "m"`
Put `"y"/"x" = "m"` in equation (1)
Comparing the equation km2 + 4m + 3 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = k , 2h = 4, b = 3
m1 = 3m2 ...(given condition)
m1 + m2 = `"-2h"/"k" = -4/"k"`
m1m2 = `"a"/"b" = 3/"k"`
m1 + m2 = `-4/"k"`
4m2 = `-4/"k"` .....(m1 = 3m2)
m2 = `-1/"k"`
m1m2 =`3/"k"`
`3"m"_2^2 = 3/"k"` .......(m1 = 3m2)
`3(-1/"k")^2 = 3/"k"` ......(m2 = `-1/"k"`)
`1/"k"^2 = 1/"k"`
`"k"^2 = "k"`
k = 1 or k = 0
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