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Question
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is three times the other, prove that 3h2 = 4ab.
Solution
Let m1 and m2 be the slopes of the lines represented by ax2 + 2hxy + by2 = 0
We are given that m2 = 3m1
∴ `"m"_1 + 3"m"_1 = - "2h"/"b"`
∴ `4"m"_1 = - "2h"/"b"`
∴ `"m"_1 = - "h"/"2b"`
Also, m1(3m1) = `"a"/"b"`
∴ `3"m"_1^2 = "a"/"b"`
∴ `3 (- "h"/"2b")^2 = "a"/"b"` ....[By (1)]
∴ `"3h"^2/"4b"^2 = "a"/"b"`
∴ 3h2 = 4ab, as b ≠ 0
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