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If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab. - Mathematics and Statistics

Sum

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab.

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Solution

Let m1 and m2 be the slopes of the lines given by ax2 + 2hxy + by2 = 0 

∴ m1 + m2 = `- "2h"/"b"` 

and m1m2 = `"a"/"b"`

We are given that m2 = 4m1

∴ m1 + 4m1 = `- "2h"/"b"`

∴ 5m1 = `(- "2h")/"b"`

∴ m1 = `- "2h"/"5b"`    .....(1)

Also, m1(4m1) = `"a"/"b"`

∴ `4"m"_1^2 = "a"/"b"`

∴ `"m"_1^2 = "a"/"4b"`

∴ `("- 2h"/"5b")^2 = "a"/"4b"`     ...[By(1)]

∴ `"4h"^2/"25b"^2 = "a"/"4b"`

∴ `"4h"^2/"25b" = "a"/4`, as b ≠ 0

∴ 16h2 = 25ab

This is the required condition.

Concept: Homogeneous Equation of Degree Two
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