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Question
Find k, if one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.
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Solution
The auxiliary equation of the lines given by 3x2 - kxy + 5y2 = 0 is 5m2 - km + 3 = 0
Now, one line is perpendicular to the line 5x + 3y = 0, whose slope is `- 5/3`
∴ slope of that line = m = `3/5`
∴ m = `3/5` is the root of the auxiliary equation 5m2 - km + 3 = 0
∴ `5(3/5)^2 - "k"(3/5) + 3 = 0`
∴ `9/5 - "3k"/5 + 3 = 0`
∴ 9 - 3k + 15 = 0
∴ 3k = 24
∴ k = 8
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